https://koki-kazaore.github.io/mathematical-optimization-papers-rss/feed.xml Latest papers on mathematical optimization from arXiv and Semantic Scholar http://www.rssboard.org/rss-specification python-feedgen en Wed, 06 May 2026 08:25:17 +0000 https://www.semanticscholar.org/paper/028e098db1db4bbe0da161d24eb9ac1f4474e44a No abstract available. doi:10.17771/pucrio.acad.69614 Tue, 14 Apr 2026 14:49:22 +0000 https://www.semanticscholar.org/paper/0073b5f7c09801aa9740f8f17aad511e26e56c08 Dans cette thèse on s’intéresse au problème des décompositions canoniques polyadiques de tenseurs d’ordre N potentiellement grands et sous différentes contraintes (non-négativité, aspect creux lié à une possible surestimation du rang du tenseur). Pour traiter ce problème, nous proposons trois nouvelles approches itératives différentes: deux approches déterministes dont une approche proximale, et une approche stochastique. La première approche étend les travaux de thèse de J-P. Royer au cas de tenseurs de dimension N. Dans l’approche stochastique, nous considérons pour la première fois dans le domaine des décompositions tensorielles, des algorithmes génétiques (mimétiques) dont principe général repose sur l’évolution d’une population de candidats. Dans le dernier type d’approche, nous avons considéré un algorithme proximal pré-conditionné (le Block-Coordinate Variable Metric Forward-Backward), algorithme fonctionnant par blocs de données avec une matrice de pré-conditionnement liée à chaque bloc et fondé sur deux étapes successives principales : une étape de gradient et une étape proximale. Finalement, les différentes méthodes suggérées sont comparées entre elles et avec d’autres algorithmes classiques de la littérature sur des données synthétiques (à la fois aléatoires ou proches des données observées en spectroscopie de fluorescence) et sur des données expérimentales réelles correspondant à une campagne de surveillance des eaux d’une rivière et visant à la détection d’apparition de polluants. doi:10.70675/3459c071zc780z47ffz86b0z9386f680ab57 Tue, 14 Apr 2026 14:49:22 +0000 https://www.semanticscholar.org/paper/00809c1a4526f981d3e462ac7a331e736f90bc83 : With e-commerce expanding rapidly, last-mile delivery challenges have been exacerbated, necessitating innovative logistics to reduce operational costs and improve delivery speed. This paper investigates a traveling salesman problem with drone stations, where a truck collaborates with multiple drones docked at candidate drone stations to serve customers. In contrast to existing studies that typically assume fixed drone speeds, this work treats drone speeds as decision variables and introduces a comprehensive energy consumption model that accounts for all phases of drone flight. The objective is to jointly optimize truck routing, station selection, drone – customer assignment, and drone speed to minimize the total delivery cost. Through a speed-discretion method, we formulate the problem as a mixed-integer linear programming model and develop a tailored adaptive large neighborhood search (ALNS) algorithm. Computational experiments indicate that for large-sized instances with 80 – 100 customers and 16 – 20 candidate stations, ALNS produces solutions within 50 seconds, with average optimality gaps below 1.8% compared to Gurobi’s solutions obtained under a 5000-second time limit. The results also show that the speed optimization strategy consistently outperforms fixed-speed approaches across multiple performance metrics, including total cost, service completion time, energy consumption, and service coverage. doi:10.1016/j.cie.2026.111941 Tue, 14 Apr 2026 14:49:22 +0000 https://www.semanticscholar.org/paper/007454fec21427f100d7d1113d65e91e6fb1e899 No abstract available. title:taxefficient retirement withdrawal planning using a linear programming model|unknown|2026 Tue, 14 Apr 2026 14:49:22 +0000 https://www.semanticscholar.org/paper/00739fd8ffff803d31892d327abc62029f9cfac2 Objectives: In silico clinical trials are a well-accepted model-informed drug development (MIDD) tool to understand, optimize and predict the effect of drugs across diverse populations [1,2]. Generation and selection of virtual patients for simulating in silico trials remains an open area of research not least due to challenges of selecting virtual patients that are representative of the underlying physiological and clinical characteristics of individual subjects enrolled into clinical trials [3]. While virtual patients can be obtained by sampling sets of parameters from a calibrated non-linear mixed effects model, these may however generate biomarkers outside the expected range of observations, thus impacting the accuracy of in silico clinical trials predictions. To obtain a more representative virtual population, different algorithms can be applied to tailor plausible patient cohorts and yield calibrated virtual patients. In this work, we present a new virtual patient selection workflow using a minimal quantitative systems pharmacology (QSP) model of chronic hepatitis B virus (HBV) infection. Methods: A QSP model of HBV disease progression incorporating the effect of standard-of-care therapies (peg-interferon and nucleos(t)ide analogues) was used to generate plausible patient cohorts corresponding to the Everest trial [4] (a real-world clinical study exploring how interferon therapy can achieve functional cure in chronically infected patients). To tailor plausible patients to a virtual cohort mimicking baseline characteristics of patients enrolled into the Everest project, we developed a computational algorithm based on a mixed-integer linear programming framework coupled to a multi-objective optimization genetic algorithm. Subsequently, the virtual patient cohort obtained with this algorithm was used to perform in silico clinical trials mirroring the Everest protocol to validate the workflow. Non-linear relationships between baseline characteristics, prognostic biomarkers, and dosing parameters with clinical endpoints were explored using in silico trials. Results: We show that: (1) the HBV QSP model captures longitudinal biomarker data from untreated patients as well as chronically infected subjects receiving standard-of-care therapies, (2) Plausible patients generated with the model capture HBV disease progression, (3) The novel computational workflow efficiently selects virtual patients matching HBsAg and viral load baseline Everest data distributions, (4) In silico trials that mirrored the Everest trial showed quantitative agreement with clinical end-points thus validating the workflow, (5) In silico trials were leveraged to identify and compare optimal dosing regimens and explore mechanistic pathways in responders and non-responders to interferon. Conclusions: This work proposes and validates a novel QSP-based computational workflow for performing in silico trials, generating mechanistic hypotheses, and identifying optimal dosing regimens.Citations: COI: DW, AN, AS, RD are employees of and hold shares in GSK. JCR is a joint postdoc with WJJ (University of Buffalo) and GSK.[1] Madabushi et al. Pharm Res. 2022[2] Rieger et al. Prog Biophys Mol Biol. 2018[3] Arsène et al. NY: Springer US. 2023[4] Xie et al. Int Liver Meeting. 2022 doi:10.70534/kzql1879 Tue, 14 Apr 2026 14:49:22 +0000 https://www.semanticscholar.org/paper/005368d90a35e2d353ab0d0b1196f2b54e9f955f No abstract available. title:optimization of investments in cybersecurity a linear optimization of investment|swati jain|2026 Tue, 14 Apr 2026 14:49:22 +0000 https://www.semanticscholar.org/paper/004c2cbb72b8e42cf48478bdca721a1985d3d5de For modeling imprecise and indeterminate data for multi-objective decision making, two different methods: neutrosophic multi-objective linear/non-linear programming neutrosophic goal programming, which have been very recently proposed in the literatuire. In many economic problems, the well-known probabilities or fuzzy solutions procedures are not suitable because they cannot deal the situation when indeterminacy inherently involves in the problem. In this case we propose a new concept in optimization problem under uncertainty and indeterminacy. It is an extension of fuzzy and intuitionistic fuzzy optimization in which the degrees of indeterminacy and falsity (rejection) of objectives and constraints are simultaneously considered together with the degrees of truth membership (satisfaction/acceptance). The drawbacks of the existing neutrosophic optimization models have been presented and new framework of multi-objective optimization in neutrosophic environment has been proposed. The essence of the proposed approach is that it is capable of dealing with indeterminacy and falsity simultaneously. title:g lobal|surapati pramanik|2026 Tue, 14 Apr 2026 14:49:22 +0000 https://www.semanticscholar.org/paper/019a75f1df24f379a909ed29aab610ac7c584038 No abstract available. title:quantuminspired hamiltonian descent for mixedinteger quadratic programming|shreya chaudhary|2026 Tue, 14 Apr 2026 14:57:01 +0000 http://arxiv.org/abs/2604.27518v1 This paper presents lpviz, a browser-based visualization tool for linear programming. lpviz is deeply interactive, offering an intuitive interface where users can directly draw and edit the feasible region and objective vector, without requiring cumbersome manipulation of raw numerical coefficients. lpviz lets users compare the behavior of several classes of linear programming algorithms, namely Simplex, Interior-Point, Primal-Dual Hybrid Gradient, and Central Path. In the 3D mode, lpviz places iterates at heights corresponding to important solver metadata such as complementarity gap or KKT residual, helping users gain further insight into algorithm behavior beyond the primal iterates alone. lpviz has been used in both research and classroom settings, to help develop intuition for the strengths and weaknesses of different solvers and the impact of solver settings on convergence behavior. lpviz is open-source, permissively licensed, and freely available on any device with a web browser at https://lpviz.net . arxiv:2604.27518 Fri, 01 May 2026 08:10:31 +0000 http://arxiv.org/abs/2604.27587v1 This paper develops a sliding mode control based frame work for equality constrained optimization by reformulation the first order Karush Kuhn Tucker conditions as control affine dynamical system. The optimization variables are treated as states and the Lagrange multipliers as control input, with equality constraints defined as sliding manifold. The resulting design guarantees exact constraint enforcement with finite time convergence, independent of objective convexity, and exhibits robustness to matched disturbance, structural uncertainty and bounded measurement noise. To accelerate the convergence, a nonsingular terminal sliding mode based normed gradient flow is introduced, ensuring both finite time convergence to optimal solution and constraint satisfaction. Rigorous Lyapunov analysis establishes closed loop stability and convergence. Numerical studies across diverse benchmark problems demonstrate superior accuracy and robustness over classical continuous time optimization method, highlighting effectiveness under disturbance. arxiv:2604.27587 Fri, 01 May 2026 08:10:31 +0000 http://arxiv.org/abs/2604.27610v1 In finance, portfolio management is a traditional yet difficult problem that has drawn attention from practitioners and researchers for many years. However, there are still difficult technological problems that need to be solved. In the world of finance, managing a portfolio has never been easy. Selecting portfolios in a volatile market is made easier with the help of portfolio management. The goal of this review study is to present the concept of physics-informed neural networks because they provide a novel approach to directly incorporating physics and finance principles into the neural network's learning process. By doing so, physics-informed neural networks ensure that their forecasts are in line with established financial regulations and processes in addition to offering precise forecasts. Furthermore, this article provides an overview of the current state of research in portfolio optimization with the support of mathematical models, deep learning models and physics-informed neural networks. In addition, the advantages and disadvantages of various deep learning and mathematical modelling are discussed. Researchers and business professionals alike should find the data useful for advancing the field of investment management and trying out new portfolio management strategies. For this purpose, in this review work, emphasis is given to these factors. Finally, a few challenging issues and potential future directions are discussed, encouraging readers to consider fresh ideas in this field of study. arxiv:2604.27610 Fri, 01 May 2026 08:10:31 +0000 http://arxiv.org/abs/2604.27622v1 Order picking travel dominates much of warehouse effort, and exact routing is especially valuable when storage is scattered so pick locations are not fixed in advance. We address the single picker routing problem (SPRP) and its scattered-storage variant (SPRP-SS) in single-block and two-block rectangular warehouses. We propose two mixed-integer linear programming formulations that exploit structural properties of optimal tours to simplify connectivity modelling and remove redundant edge configurations: a Configuration Connectivity model tailored to single-block layouts and an Edge Connectivity model that extends to two-block layouts. In extensive computational experiments on large randomly generated benchmark sets for single-block and two-block rectangular layouts, we compare these formulations against established MILP and network-flow baselines for SPRP and SPRP-SS and report computational gains tied to the structural restrictions. The results support using compact, solver-based exact routing models in industrial settings where dynamic programming is cumbersome to integrate, particularly for SPRP-SS and for routing subproblems embedded in larger planning or warehouse-design optimizations. arxiv:2604.27622 Fri, 01 May 2026 08:10:31 +0000 http://arxiv.org/abs/2604.27705v1 This paper considers the robust control of a catenary robot composed of two quadrotors connected by an inextensible cable. The system is modeled on \(SE(3)\), with the cable treated as a geometric subsystem induced by the UAV configuration rather than as an independent dynamical element. The catenary shape determines configuration-dependent forces that couple the translational dynamics of the vehicles. We propose a geometric tracking controller for the relative configuration of the agents and analyze its robustness with respect to unstructured uncertainties in the catenary-induced forces. The main theoretical result establishes local input-to-state stability of the closed-loop tracking errors. In particular, we obtain asymptotic convergence in the nominal case and an explicit ultimate bound for the tracking errors under bounded catenary-force perturbations. arxiv:2604.27705 Fri, 01 May 2026 08:10:31 +0000 http://arxiv.org/abs/2604.27755v1 We introduce Gårding polynomials, a class of real multivariate polynomials defined via positivity regions invariant under translation by positive directions and closed under strictly positive affine transformations. We establish a structural theorem providing two complementary characterizations of this class: one via reduction to the multi-affine case through polarization, and another via a recursive condition involving partial derivatives. The class of Gårding polynomials strictly extends that of real stable polynomials while retaining many of their structural properties. In particular, multi-affine Gårding polynomials with nonnegative coefficients satisfy the Rayleigh property, and their positive univariate specializations yield ultra log-concave coefficient sequences. Moreover, the Gårding property for several matroid generating functions is preserved under natural matroid operations. As applications, we obtain new negative dependence results for generating functions associated with various classes of matroids and graphs--many of which lie beyond the reach of real stability or Lorentzian methods--as well as for characteristic polynomials of certain matrix classes. arxiv:2604.27755 Fri, 01 May 2026 08:10:31 +0000 http://arxiv.org/abs/2604.27922v1 This paper studies data-driven approaches to the continuous-time linear quadratic regulator (LQR) problem based on two existing parameterizations, namely a closed-loop (CL) parameterization from behavioral system theory and an integral reinforcement learning (IRL) parameterization. The CL parameterization characterizes the closed-loop system via a matrix that satisfies equality constraints. While this parameterization has been extensively studied for discrete-time systems, we adapt key results to the continuous-time setting and develop a policy iteration (PI) scheme, derive a data-driven continuous-time algebraic Riccati equation (CARE), and introduce an alternative convex problem formulation. The IRL parameterization utilizes off-policy data to perform policy evaluation, which is then used for PI or value iteration. Within the IRL framework, we derive a policy gradient flow and propose convex reformulations of the LQR problem. Finally, we provide a unified treatment of these parameterizations that enables a systematic understanding of existing approaches and clarifies their structural relationships. arxiv:2604.27922 Fri, 01 May 2026 08:10:31 +0000 http://arxiv.org/abs/2604.28006v2 We consider smooth convex minimization over compact convex sets, i.e., $\min_{x \in C} f(x)$ with the (vanilla) Frank-Wolfe algorithm. Well-known lower bounds establish a worst-case $Ω(1/t)$ primal-gap barrier in the general smooth convex case, and faster convergence usually requires favorable function properties such as Hölder error bounds or strong convexity. We present a new Local Dual Sharpness (LDS) condition, essentially a property of the feasible region and its LMO, under which the Frank-Wolfe algorithm converges in $o(1/t)$ for any smooth convex function, ruling out an $Ω(1/t)$ lower bound under LDS. The condition is a generalization (and localization) of uniform convexity of sets and it is satisfied by any uniformly convex set. To our knowledge, this is the first unconditional $o(1/t)$ convergence result for uniformly convex sets. Combining LDS with stronger function properties, e.g., a local variant of Hölder error bounds, allows us to quantify the actual rates. arxiv:2604.28006 Fri, 01 May 2026 08:10:31 +0000 http://arxiv.org/abs/2604.28110v1 In this paper, we propose a scaled gradient modified non-monotone line search method for solving constrained minimization problems, and explore several specific properties of this method, namely, its convergence analysis. We discuss the linear convergence rate of the sequence generated by the proposed algorithm to a solution of the constrained minimization problem where the objective function is strongly quasiconvex. We consider numerical examples of large-scale fractional programming and quadratic programming for the function of pseudo convex and strongly quasiconvex and compare the performance of the proposed algorithm with the existing ones for these examples. arxiv:2604.28110 Fri, 01 May 2026 08:10:31 +0000 http://arxiv.org/abs/2604.28144v1 Efficient exploration is a central problem in reinforcement learning and is often formalized as maximizing the entropy of the state-action occupancy measure. While unconstrained maximum-entropy exploration is relatively well understood, real-world exploration is often constrained by safety, resource, or imitation requirements. This constrained setting is particularly challenging because entropy maximization lacks additive structure, rendering Bellman-equation-based methods inapplicable. Moreover, scalable approaches require policy parameterization, inducing non-convexity in both the objective and the constraints. To our knowledge, the only prior model-free policy-gradient approach for this setting under general policy parameterization is due to Ying et al. (2025). Unfortunately, their guarantees are limited to weak regret and ergodic averages, which do not imply that the final output is a single deployable policy that is near-optimal and nearly feasible. In this work we take a different approach to this problem, and propose Policy Gradient Penalty (PGP) method, a single-loop policy-space method that enforces general convex occupancy-measure constraints via quadratic-penalty regularization. PGP constructs pseudo-rewards that yield gradient estimates of the penalized objective, subsequently exploiting the classical Policy Gradient Theorem. We further establish the regularity of the penalized objective, providing the smoothness properties needed to justify the convergence of PGP. Leveraging hidden convexity and strong duality, we then establish global last-iterate convergence guarantees, attaining an $ε$-optimal constrained entropy value with $ε$ bounded constraint violation despite policy-induced non-convexity. We validate PGP through ablations on a grid-world benchmark and further demonstrate scalability on two challenging continuous-control tasks. arxiv:2604.28144 Fri, 01 May 2026 08:10:31 +0000 https://www.semanticscholar.org/paper/00c9db6ce1fe01eae4bff4fcd7934702395f1137 Causal inference is essential for data-driven decision-making, as it aims to uncover causal relationships from observational data. However, identifying causality remains challenging due to the potential for confounding and the distinction between correlation and causation. While recent advances in causal machine learning and matching algorithms have improved estimation accuracy, these methods often face trade-offs between interpretability and computational efficiency. This paper proposes a novel approach that combines a tree-based discretization technique, tailored for causal inference, with an integer linear programming-based matching algorithm. The discretization ensures approximately linear relationships for control datasets within strata, enabling effective matching, while the optimization framework optimizes for global balance. The resulting algorithm yields computational efficiency and less biased ATT estimates compared to state-of-the-art algorithms. Empirical evaluations demonstrate the proposed method's practical advantages over existing techniques in causal inference scenarios. arxiv:2604.27307 Sun, 03 May 2026 07:59:48 +0000 http://arxiv.org/abs/2605.00131v2 This paper studies a variant of the Set Covering Routing Problem (SCRP) motivated by post-disaster humanitarian logistics. We consider a hybrid distribution concept in which the majority of transportation is performed by helicopters, while ground transport is limited to the last mile, addressing severe accessibility constraints in disaster-affected regions. The resulting problem integrates landing site location, routing, and covering decisions, incorporating features of the Multi-Vehicle Covering Tour Problem (m-CTP) and the Vehicle Routing with Demand Allocation Problem (VRDAP) in a facility-capacitated, multi-depot setting. Due to the computational complexity of the problem, we develop an Adaptive Variable Neighborhood Search (AVNS) that combines established routing operators with novel mechanisms for covering decisions. The performance of the proposed approach is evaluated on benchmark instances for the related m-CTP and VRDAP problems, demonstrating competitive solution quality compared to problem-specific state-of-the-art approaches. Furthermore, we apply our AVNS to a real-world case study based on the 2024 flash floods in Afghanistan. The results highlight the practical relevance of the proposed framework and provide managerial insights into effective distribution strategies for disaster response operations. arxiv:2605.00131 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00155v1 Reinforcement learning from human feedback (RLHF) has become a core post-training step for aligning large language models, yet the reward signal used in RLHF is only a learned proxy for true human utility. From an operations research perspective, this creates a decision problem under objective misspecification: the policy is optimized against an estimated reward, while deployment performance is determined by an unobserved objective. The resulting gap leads to reward over-optimization, or Goodharting, where proxy reward continues to improve even after true quality deteriorates. Existing mitigations address this problem through uncertainty penalties, pessimistic rewards, or conservative constraints, but they can be computationally burdensome and overly pessimistic. We propose Wasserstein distributionally robust regret optimization (DRRO) for RLHF. Instead of pessimizing worst-case value as in standard DRO, DRRO pessimizes worst-case regret relative to the best policy under the same plausible reward perturbation. We study the promptwise problem through a simplex allocation model and show that, under an $\ell_1$ ambiguity set, the inner worst-case regret admits an exact solution and the optimal policy has a water-filling structure. These results lead to a practical policy-gradient algorithm with a simple sampled-bonus interpretation and only minor changes to PPO/GRPO-style RLHF training. The framework also clarifies theoretically why DRRO is less pessimistic than DRO, and our experiments show that DRRO mitigates over-optimization more effectively than existing baselines while standard DRO is systematically over-pessimistic. arxiv:2605.00155 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00158v1 Many incentive design problems must contend with information asymmetries due to non-observation of efficiency (adverse selection) or non-observation of effort (moral hazard). And although a growing body of literature considers incentive design in control systems, the problem of designing incentives for control systems under information asymmetries has been less well-studied. This paper considers a model of moral hazard within control systems. In our model, the control system is described by an (affine) linear time-invariant (LTI) system with process noise. There is an agent who gets to choose (from between two choices) a linear state-feedback controller to apply to the LTI system, with one of the state-feedback controllers having a higher quadratic cost on the control inputs than the other. Our goal is to design a payment scheme that incentivizes the agent to choose the state-feedback controller that minimizes a quadratic cost on system states plus the time-discounted payment amount, subject to the understanding that the agent bears the control cost while being risk-averse with respect to their time-discounted payment. We formulate the problem as a constrained optimization, and prove that for a payment given after a fixed (but optimizable) time horizon the optimal payment scheme chooses the payment amount using a likelihood ratio hypothesis test. We numerically demonstrate our results by applying the derived optimal payment scheme to two examples: load frequency control (LFC) in power systems and wellness interventions for body weight loss. arxiv:2605.00158 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00160v1 Decentralized stochastic control problems are difficult to study due to information structure dependent subtleties, which prevent many classical methods in stochastic control from being applicable. In this paper we consider such problems with general standard Borel spaces under two related information structures. (a) the one-step delayed information sharing pattern (OSDISP) where agents share their information with one-step delay, and (b) the $K$-step periodic information sharing pattern (KSPISP), where information is shared periodically. It is known that OSDISP and KSPISP problems admit a centralized reduction where the agents view the problem from the perspective of a centralized controller that uses the common information to prescribe function valued actions (local policies) which map each agent's private information to an optimal action in the original problem. We provide rigorous approximation results and performance bounds for the KSPISP and OSDISP problems, which results from replacing the full common information by a finite sliding window of information and we establish near optimality of such policies. The latter depends on a predictor stability condition in expected total variation. As a further contribution, we show that under the information structures provided, corresponding Q-learning algorithms (in quantized or finite memory forms) converge asymptotically to near optimal solutions. While restrictive and hypothetical conditions have been presented in the literature, our contributions are thus to provide, to our knowledge, the first explicit conditions and rigorous approximation and learning results for such decentralized problems with general spaces. arxiv:2605.00160 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00212v1 We develop a structure-preserving solution framework for the optimal control of the time-dependent Maxwell's equations. Building on a well-posedness theory for a weak form of the forward problem, we first analyze a forward solver that couples Nédélec and Raviart--Thomas finite elements with Crank--Nicolson time stepping. The solver preserves the de~Rham structure, enforces a discrete Gauss law, exactly satisfies a per-time-step energy balance, and converges to the weak solution under low regularity assumptions on the problem data, which are dictated by the optimal control setting. To control the Maxwell system, we add the curl of a space-time current density as a source to Ampére's law. The curl form yields charge conservation without auxiliary constraints. We prove the well-posedness and continuity of the control-to-state map, derive the adjoint system and a gradient representation for a tracking-type objective functional, and formulate a discrete optimization scheme that inherits structure preservation from the forward solver. Our discrete stationarity conditions are consistent with their continuous counterparts, and the discrete optimal controls converge, with mesh and time refinements, to the continuous optima. We demonstrate the merits of our optimal control formulation and the theoretical developments by numerically solving a series of source-cloaking model problems. arxiv:2605.00212 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00229v1 We study the problem of training diffusion and flow generative models to sample from target distributions defined by an exponential tilting of a base density; a formulation that subsumes both sampling from unnormalized densities and reward fine-tuning of pre-trained models. This problem can be approached from a stochastic optimal control (SOC) perspective, using adjoint-based or score matching methods, or from a non-equilibrium thermodynamics perspective. We provide a unified framework encompassing these approaches and make three main contributions: (i) bias-variance decompositions revealing that Adjoint Matching/Sampling and Novel Score Matching have finite gradient variance, while Target and Conditional Score Matching do not; (ii) norm bounds on the lean adjoint ODE that theoretically support the effectiveness of adjoint-based methods; and (iii) adaptations of the CMCD and NETS loss functions, along with novel Crooks and Jarzynski identities, to the exponential tilting setting. We validate our analysis with reward fine-tuning experiments on Stable Diffusion 1.5 and 3. arxiv:2605.00229 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00281v1 We study high-probability (HP) convergence guarantees in decentralized stochastic optimization, where multiple agents collaborate to jointly train a model over a network. Existing HP results in decentralized settings almost exclusively focus on the Decentralized Stochastic Gradient Descent ($\mathtt{DSGD}$) algorithm, which requires strong assumptions, such as bounded data heterogeneity, or strong convexity of each agent's cost. This is contrary to the mean-squared error (MSE) results, where methods incorporating bias-correction techniques are known to converge under relaxed assumptions and achieve better practical performance. In this paper we provide the first step toward bridging the gap, by studying HP convergence of $\mathtt{DSGD}$ incorporating the gradient tracking technique, in the presence of noise satisfying a relaxed sub-Gaussian condition. We show that the resulting method, dubbed $\mathtt{GT-DSGD}$, achieves order-optimal HP convergence rates for both non-convex and Polyak-Łojasiewicz costs, of order $\mathcal{O}\Big(\frac{\log(1/δ)}{\sqrt{nT}}\Big)$ and $\mathcal{O}\Big(\frac{\log(1/δ)}{nT}\Big)$, respectively, where $n$ is the number of agents, $T$ is the time horizon and $δ\in (0,1)$ is the confidence parameter. Our results establish that $\mathtt{GT-DSGD}$ converges in the HP sense under the same conditions on the cost as in the MSE sense, while achieving comparable transient times. To the best of our knowledge, these are the first HP guarantees for decentralized optimization methods incorporating bias-correction. Numerical experiments on real and synthetic data verify our theoretical findings, underlining the superior performance of $\mathtt{GT-DSGD}$ and highlighting that the benefits of incorporating bias-correction are also maintained in the HP sense. arxiv:2605.00281 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00298v1 Deploying reinforcement learning policies in the real world requires adapting to time-varying environments. We study this problem in the contextual Markov Decision Process (cMDP) framework, where a family of environments is indexed by a low-dimensional context unknown at test time. The standard approach decomposes the problem: train a so-called "universal policy" which assumes knowledge of the true context, then pair it with a context estimator which approximates context using the observed trajectory. We identify a simple, counterintuitive trick that substantially improves the estimator: randomly delete a fraction of the training buffer after each round. This works because data is collected across multiple rounds using progressively better policies, and older trajectories come from a different distribution than what the estimator will face at deployment time; random deletion creates an implicit exponential decay on older data while preserving diversity without requiring any explicit identification of which samples are stale. This reduces robustness gap by 30% for MLPs and by 6% on average for recurrent networks. Strikingly, it allows a narrow MLP with 5x fewer parameters to outperform a wide MLP trained without deletion. To understand when and why deletion helps, we analyze regularized empirical risk minimization with a mismatch between the train distribution and the distribution at deployment; in this idealized setting, we prove that removing a single uniformly random training point decreases expected test loss in expectation under mild conditions. For ridge regression we make this quantitative: deletion helps when the regularization coefficient is moderate and the signal-to-noise ratio (SNR) is sufficiently low, and, crucially, this SNR threshold gives a direct measure of how large the distribution mismatch between training and deployment must be for deletion to be beneficial. arxiv:2605.00298 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00311v1 This paper presents a unified derivation of transversality conditions in optimal control problems using exact penalty functions. The key regularity condition is that the origin is uniformly separated from the subdifferential of the penalty function in a neighborhood of the admissible set. This condition, hereafter referred to as the Unified Separation Condition (USC), generalizes the classical Mangasarian-Fromovitz condition for inequalities and linear independence of gradients for equalities; in the smooth case, these classical conditions are equivalent to USC, as shown via Gordan's theorem. The USC remains applicable even when constraint functions are nondifferentiable, where classical constraint qualifications are not defined. Assuming exactness, we derive transversality conditions for all major cases: fixed and free terminal time, equality and inequality constraints, moving manifolds, and free left endpoint. Remarkably, this approach yields these classical results in a concise and transparent manner, avoiding the need for constructing cones of endpoint variations or applying separation theorems. The theoretical results are complemented by a numerical implementation applied to the time-optimal control of a harmonic oscillator. The numerical implementation converges to the exact solution obtained via Pontryagin's maximum principle combined with transversality conditions, confirming the consistency and practical applicability of the proposed methodology. arxiv:2605.00311 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00386v1 Our aim is to explain mathematical programs with equilibrium constraints (MPECs), motivate them through applications, present the main equivalent formulations of equilibrium constraints, and summarize the basic existence theory for optimal solutions. The central message is that an MPEC is an optimization problem whose feasible set is partly defined by another optimization, variational inequality, complementarity system, or equilibrium model. arxiv:2605.00386 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00387v1 We present a focused introduction to exact penalty methods for nonlinear programs and mathematical programs with equilibrium constraints (MPECs), emphasizing their connection to modern error bound theory. The goal is twofold. First, we explain how classical optimality conditions can be interpreted through exact penalization, and why such results typically rely on constraint regularity conditions that can be understood as error bounds on perturbations of feasible sets. We then highlight how recent developments based on subanalytic geometry and Lojasiewicz-type inequalities extend this framework beyond classical regularity assumptions, enabling exact penalization under broader analytic conditions. Second, we demonstrate how this theory can be applied in practice to MPECs by reformulating them via KKT systems and constructing exact penalty functions based on residual mappings. Particular attention is given to fractional-order penalties arising from Lojasiewicz error bounds, as well as to improved formulations for special problem classes where sharper exponents can be obtained. These developments provide both theoretical insight and practical guidance for analyzing and solving challenging constrained optimization problems. arxiv:2605.00387 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00388v1 We present a systematic introduction to first-order optimality conditions for mathematical programs with equilibrium constraints (MPECs), emphasizing the limitations of classical nonlinear programming techniques. The goal is twofold. First, we explain why a direct application of standard optimality conditions -- based on reformulating MPECs via KKT systems or differentiable exact penalty functions -- is often inadequate, as such approaches typically require strong and restrictive assumptions, including nondegeneracy and smoothness conditions. Second, we develop a first-principles framework for analyzing MPECs by focusing on the geometric structure of the feasible region. In particular, we study stationarity concepts and provide a detailed characterization of the tangent cone at feasible points, which leads to appropriate constraint qualifications tailored to MPECs. These results form the foundation for rigorous first-order analysis and clarify the relationship between the original MPEC formulation and its KKT-based representation, offering practical guidance for handling these inherently challenging optimization problems. arxiv:2605.00388 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00395v1 Classical flocking models demonstrate how local interactions generate emergent order, but real-world multi-agent deployments are bound by severe constraints: limited actuator availability, heterogeneous communication latencies, and environmental noise. In this talk, we present a unified finite-N framework that tackles the interplay of these exact mechanisms. We study a delayed stochastic leader-follower particle system featuring topological communication, singular repulsion, and bounded sparse leader actuation. A central challenge in such systems is mathematical well-posedness, as discontinuous communication laws and singular repulsions clash with standard strong Ito frameworks. We resolve this by introducing an augmented Lyapunov functional that simultaneously enforces a strict collision barrier and closes a uniform Gronwall estimate. Building on this rigorous foundation, we formulate a free-terminal-time, chance-constrained optimal control problem. We show that temporally sparse, bang-off-bang leader actuation not only drastically reduces control effort compared to continuous baselines, but also reveals non-monotone sensitivities to leader density. Ultimately, we demonstrate that in delayed stochastic swarms, adding more direct actuation is not strictly optimal -- highlighting a highly non-trivial resource allocation paradox in cooperative control. arxiv:2605.00395 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00396v1 In Riemannian optimization, it is well known that the condition number of the Riemannian Hessian at an optimum strongly influences the asymptotic convergence behavior of optimization algorithms. On the manifold of symmetric positive definite (SPD) matrices, several commonly used metrics for optimization, such as the Affine-Invariant (AI) and Bures--Wasserstein (BW) metrics, tend to become ill-conditioned as the underlying SPD matrix becomes ill-conditioned. As a result, even when the Euclidean Hessian remains uniformly well-conditioned on the SPD manifold, optimization may still become difficult near an optimum associated with an ill-conditioned SPD matrix. In this paper, we address this issue through the Alpha-Procrustes (AP) geometry on the SPD manifold. This geometry generalizes several well-known metrics, including the Log-Euclidean (LE) metric for \(α=0\) and the BW metric for \(α=1/2\). We first show that, when \(α=1\), all eigenvalues of the Riemannian metric operator induced by the AP geometry are uniformly bounded independently of the underlying SPD matrix. Therefore, under the assumption that the Euclidean Hessian satisfies the uniform spectral bounds, all the eigenvalues of the corresponding Riemannian Hessian are uniformly bounded independently of the underlying SPD matrix. Consequently, the case \(α=1\) provides a robust geometric framework for several Riemannian optimization problems involving ill-conditioned SPD matrices. Finally, we validate our theoretical findings through extensive numerical experiments across a range of applications. arxiv:2605.00396 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00473v1 Multi-task learning (MTL) has emerged as a pivotal paradigm in machine learning by leveraging shared structures across multiple related tasks. Despite its empirical success, the development of likelihood-based efficiently solvable algorithms--even for shared linear representations--remains largely underdeveloped, primarily due to the non-convex structure intrinsic to matrix factorization. This paper introduces a first-order algorithm that jointly learns a shared representation and task-specific parameters, with guaranteed efficiency. Notably, it converges in $\widetilde{\mathcal{O}}(1)$ iterations and attains a \emph{near-optimal} estimation error of $\widetilde{\mathcal{O}}(dk/(TN))$, \emph{improving} over existing likelihood-based methods by a factor of $k$, where $d$, $k$, $T$, $N$ denote input dimension, representation dimension, task count, and samples per task, respectively. Our results justify that likelihood-based first-order methods can efficiently solve the MTL problem. arxiv:2605.00473 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00532v1 We establish a connection between policy evaluation in Markov decision processes and PageRank in network analysis. For a fixed policy, we show that the value function of a discounted Markov decision process can be obtained, up to an explicit rescaling, from the PageRank vector of a suitably defined time-reversed Markov chain. In this correspondence, the discount factor plays the role of the teleportation parameter, while rewards induce the restart distribution. Beyond the irreducible case, invoking quasi-stationary distributions and Doob $h$-transforms, we prove a general decomposition theorem showing that policy evaluation for arbitrary finite MDPs reduces to a collection of PageRank problems on the recurrent and transient components of the policy-induced Markov chain. This framework naturally extends to undiscounted MDPs with terminal states and to transition-dependent rewards. We conclude by showing efficiency of our approach on a numerical example of a sticky random walk on large deterministic and random graphs. arxiv:2605.00532 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00572v1 Algorithm performance in combinatorial optimization is highly sensitive to parameter settings, while a single globally tuned configuration often fails to exploit the heterogeneity of instances. This limitation is particularly evident in the Electric Capacitated Vehicle Routing Problem, where instances differ in structure, demand patterns, and energy constraints. This paper investigates instance-aware parameter configuration for Bilevel Late Acceptance Hill Climbing, a state-of-the-art metaheuristic for the Electric Capacitated Vehicle Routing Problem. An offline tuning procedure is used to obtain instance-specific parameter labels, which are then mapped from instance features via a regression model to enable parameter prediction for unseen instances prior to execution. Experimental results on the IEEE WCCI 2020 benchmark and its extensions show that the proposed approach achieves an average objective value reduction of $0.28\%$ across eight held-out test instances relative to a globally tuned configuration. This corresponds to a significant cost reduction in multimillion-dollar transportation operations. arxiv:2605.00572 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00581v1 Gradient Boosting Decision Trees (GBDTs) dominate tabular machine learning, with modern implementations like XGBoost, LightGBM, and CatBoost being based on Newton boosting: a second-order descent step in the space of decision trees. Despite its empirical success, the global convergence of Newton boosting is poorly understood compared to first-order boosting. In this paper, we introduce Restricted Newton Descent, which studies convex optimization with Newton's method on Hilbert spaces with inexact iterates, based on the concepts of cosine angle and weak gradient edge. Within this framework, we recover Newton boosting with GBDTs and classical finite-dimensional theory as special cases. We first prove that vanilla Newton boosting achieves a linear rate of convergence for smooth, strongly convex losses that satisfy a Hessian-dominance condition. To handle general convex losses with Lipschitz Hessians, we extend a recent gradient regularized Newton scheme to the restricted weak learner setting. This scheme minimally modifies the classical algorithm by introducing an adaptive $\ell_2$-regularization term proportional to the square root of the gradient norm at each iteration. We establish a $\mathcal{O}(\frac{1}{k^2})$ rate for this scheme, thereby obtaining a globally convergent second-order GBDT algorithm with a rate matching that of first-order boosting with Nesterov momentum. In numerical experiments, we show that our scheme converges while vanilla Newton boosting may diverge. arxiv:2605.00581 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00588v1 Demand-side management (DSM) enables distribution system operators (DSOs) to steer electricity consumption through dynamic price signals or incentive mechanisms, thereby leveraging end-users' flexibility potential for delivering grid services. The resulting hierarchical interaction between the DSO and the end-users can be formulated as a Stackelberg game, where the operator dynamically sets the prices and the end-users optimally respond to them. Efficiently designing these price signals is challenging, as the users' response models are unknown or difficult to estimate. In this paper, we propose a learning-based zeroth-order algorithm for incentive design, in which the iterative update of the incentive signals is efficiently assisted by a data-driven online estimation of the users' responses. The proposed method is then proven to converge to an equilibrium tariff while allowing the DSO to estimate the decision-making problems at the user level. Moreover, the method preserves users' privacy, as the update rule of the DSO is solely based on observations of communicated end-user actions. Numerical simulations employing real-world data illustrate the efficient convergence of our learning-based proposed method, while significantly reducing the number of required interactions between the DSO and the end-users with respect to the state-of-the-art approach. arxiv:2605.00588 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00594v1 We analyze the sum-of-squares rank of unweighted instances of the Minimum Knapsack (MK) problem, i.e., minimization of $\sum_{i=1}^n x_i$ for 0/1 variables under the constraint $\sum_{i=1}^n x_i \geq q$, with $q \in \mathbb{R}$. Such instances have long served as a testbed for understanding the limitations of lift-and-project methods in Boolean optimization. For example, both the Lovász-Schrijver and Sherali-Adams hierarchies require (maximal) rank $n$ to solve them, already when $q=1/2$ is constant. The SOS hierarchy requires only \emph{sublinear} rank $O(\sqrt{n})$ to solve unweighted MK when $q=1/2$. On the other hand, when $q$ is allowed to vary with~$n$, the SOS rank of the problem may become linear. Interestingly, this is known to happen both when $q$ is large, and when $q$ is very small ($0<q \leq 2^{-n}$). This raises the question of whether we should think of hard instances of unweighted MK as being typical for the SOS hierarchy, or as a consequence of very specific choices of the threshold parameter $q$. In this paper, we address this question by showing new upper and lower bounds on the SOS rank of unweighted MK in the whole regime of the parameter $q$. For $n-q \leq O(1)$, we show that the SOS rank is constant. In contrast, when $q \leq O(1)$, a linear rank is needed if $q$ is exponentially close to an integer. As our main positive result, we show that linear rank is very rare for $q \leq O(1)$. This can be expressed in the language of smoothed analysis: after perturbing $q$ by a Gaussian with mean $0$ and variance $σ^2$, the expected SOS rank of MK is $O(\sqrt{n} \log (n/σ))$. arxiv:2605.00594 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00654v1 For a risk-averse finite-horizon Markov Decision Problem, we introduce a special class of Markov coherent risk measures, called mini-batch measures. We also define the class of multipattern risk-averse problems that generalizes the class of linear systems. We use both concepts in a feature-based $Q$-learning method with multipattern $Q$-factor approximation and we prove a high-probability regret bound of $\mathcal{O}\big(H^2 N^H \sqrt{ K}\big)$, where $H$ is the horizon, $N$ is the mini-batch size, and $K$ is the number of episodes. We also propose an economical version of the $Q$-learning method that streamlines the policy evaluation (backward) step. The theoretical results are illustrated on a stochastic assignment problem and a short-horizon multi-armed bandit problem. arxiv:2605.00654 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00688v1 This paper is concerned with portfolio selection for an investor with exponential, power, and logarithmic utility in multi-asset financial markets allowing jumps. We investigate the classical Merton's portfolio optimization problem in a Volterra stochastic environment described by a multivariate Volterra--Heston model with jumps driven by an independent Poisson random measure. Owing to the non-Markovian and non-semimartingale nature of the model, classical stochastic control techniques are not directly applicable. Instead, the problem is tackled using the martingale optimality principle by constructing a family of supermartingale processes characterized via solutions to an original Riccati backward stochastic differential equation with jumps (Riccati BSDEJ).The resulting optimal strategies for Merton's problems are derived in semi-closed form depending on the solutions to time-dependent multivariate Riccati-Volterra equations, while the optimal value is expressed using the solution to this original Riccati BSDEJ. Numerical experiments on a two-dimensional rough Heston model illustrate the impact of both path roughness and jumps components on the value function and optimal strategies in the Merton problem. arxiv:2605.00688 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00694v1 We establish the first general regularity result for constrained optimal control problems arising naturally in mathematical physics and mathematical biology. Namely, we prove that for a large class of problems of the form ``maximise $\int ψ(Θ_m)-c\int m$ where $-ΔΘ_m=mΘ_m+B(x,Θ_m)$, under the constraint $0\leq m\leq 1$ a.e.", the solution $m^*$ is bang-bang, in the sense that $m^*=χ_{E^*}$, and that $\partial E^*$ is smooth up to a $(d-2)$-dimensional subset. Moreover, we prove that the solutions to the volume constrained problem ``maximise $\int ψ(Θ_m)$ where $-ΔΘ_m=mΘ_m+B(x,Θ_m)$, under the constraint $0\leq m\leq 1$ a.e and $\int m=m_0$" are bang-bang in the sense that $m^*=χ_{E^*}$ and that, in the two-dimensional case, $\partial E^*$ is a finite union of smooth curves. This is done via reduction to an unstable free boundary problem, the regularity analysis of which was pioneered by Monneau \& Weiss and Chanillo, Kenig \& To. In our case, the free boundary is not minimising, and the laplacian of the state function is sign-changing, which creates significant difficulties, in particular regarding the non-degeneracy of blow-ups. This requires a new approach blending tools from optimal control theory, free boundary and measure theory to establish the regularity of the free boundary. arxiv:2605.00694 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00711v1 We study decentralized optimization over networks where agents cooperatively minimize a smooth (strongly) convex sum of local losses while communicating only with immediate neighbors. Prevailing decentralized methods require either centralized knowledge of global problem and network parameters for stepsize tuning--hence impractical, or costly per-iteration line-searches that demand access to local function values. We propose line-search-free, fully decentralized algorithms in which each agent adapts its stepsize using only past local iterates and gradients--with no extra function evaluations and no global tuning. The key technical ingredient is a new Lyapunov function, from which a natural adaptive stepsize rule emerges: at each iteration, each agent selects the largest stepsize that guarantees descent, based solely on a local curvature estimate built from successive gradients. The proposed algorithms enjoy strong theoretical guarantees: sublinear convergence rates for merely convex objectives and linear rates under strong convexity. Numerical experiments on standard benchmarks show consistent improvements over the state of the art, both adaptive and non-adaptive. arxiv:2605.00711 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00728v1 We obtain Sion's minimax theorem in Hadamard spaces and discuss its applications. Among other things, we study several fundamental properties of resolvents of saddle functions in Hadamard spaces. An application to the proximal point algorithm for minimax problems in Hadamard spaces are also included. arxiv:2605.00728 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.00740v1 Randomized-subspace methods reduce the cost of first-order optimization by using only low-dimensional projected-gradient information, a feature that is attractive in forward-mode automatic differentiation and communication-limited settings. While Nesterov acceleration is well understood for full-gradient and coordinate-based methods, obtaining accelerated methods for general subspace sketches that use only projected-gradient information and can improve over full-dimensional Nesterov acceleration in oracle complexity is technically nontrivial. We develop randomized-subspace Nesterov accelerated gradient methods for smooth convex and smooth strongly convex optimization under matrix smoothness and generic sketch moment assumptions. The key technical ingredient is a three-sequence formulation tailored to matrix smoothness, which recovers the corresponding classical Nesterov methods in the full-dimensional case. The resulting theory establishes accelerated oracle-complexity guarantees and makes explicit how matrix smoothness and the sketch distribution enter the complexity. It also provides a unified basis for comparing sketch families and identifying when randomized-subspace acceleration improves over full-dimensional Nesterov acceleration in oracle complexity. arxiv:2605.00740 Mon, 04 May 2026 08:28:14 +0000 http://arxiv.org/abs/2605.03841v1 Symbolic regression aims to discover interpretable equations from data, yet modern gradient-based methods fail for operators that introduce singularities or domain constraints, including division, logarithms, and square roots. As a result, Equation Learner-type models typically avoid these operators or impose restrictions, e.g. constraining denominators to prevent poles, which narrows the hypothesis class. We propose a complex weight extension of the Equation Learner that mitigates real-valued optimization pathologies by allowing optimization trajectories to bypass real-axis degeneracies. The proposed approach converges stably even when the target expression has real-domain poles, and it enables unconstrained use of operations such as logarithm and square root. We Validate the method on symbolic regression benchmarks and show it can recover singular behavior from experimental frequency response data. arxiv:2605.03841 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.00953v2 We study the problem of locating violating principal minors in structured matrix families that lie near the boundary of P-matrices and admit sparse violations under perturbation. Viewing violation search as an information acquisition problem, we show that, despite strong underlying structure, the location of a violation may be globally encoded and not accessible through local queries under a restricted interaction model. This leads to an information-theoretic bottleneck: each query reveals only vanishing information about the violating subset, so that polynomially many queries accumulate insufficient information to identify it. Using mutual information and Fano's inequality, we show that any algorithm restricted to polynomially many queries cannot recover the violating subset with constant success probability. Our analysis highlights a distinction between structure and accessibility: even highly structured problems can be computationally intractable when the information required to locate a solution is not accessible through the available queries. arxiv:2605.00953 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.00999v1 We investigate a hierarchical control problem for a one-dimensional Stefan system with localized distributed controls. The setting combines a Stackelberg strategy with a Nash equilibrium among multiple followers, yielding a multi-objective free-boundary problem. The interaction between the hierarchical control and the moving interface results in a nonlinear optimality system, and we show that the original problem reduces to the null controllability of this optimality system. Under suitable geometric conditions on the control regions, we establish a local null controllability result. The proof relies on an observability inequality for a linearized system, obtained through Carleman estimates adapted to the presence of a moving boundary. These results constitute, to the best of our knowledge, the first treatment of a Stefan system within a Stackelberg-Nash framework. arxiv:2605.00999 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01051v1 While Bellman equations for basic reach, avoid, and reach-avoid problems are well studied, the relationship between value optimality and policy optimality becomes subtle in the undiscounted infinite-horizon setting, particularly for more complicated tasks. Greedily maximizing the Q-function can produce policies that indefinitely defer task completion for reach-avoid problems, or equivalently, Until specifications, even when the value function is optimal. Building upon recent results decomposing the value function for temporal logic (TL) into a graph of constituent value functions, we construct non-Markovian policies based on state history that avoid this pathology and prove their optimality with respect to the quantitative robustness score for nested Until, Globally, and Globally-Until specifications. We further show how the Q function can serve as a safety filter for complex TL specifications, extending prior results beyond simple avoid or reach-avoid tasks. arxiv:2605.01051 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01080v1 We study a principal-agent problem with adverse selection, where the principal does not know the agent's true cost but must design a contract to optimize a specific criterion. Unlike standard screening frameworks that allow for self-selection, we assume the principal can only offer a unique contract. We show that the agent's optimization problem can be reformulated as a stochastic target problem. After characterizing the credible domain of this target problem, we show that the principal's objective can be solved as a stochastic optimal control problem with partial information and state constraints. The description of the credible domain also allows us to obtain the value of screening contracts. arxiv:2605.01080 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01178v1 We develop a stochastic game-theoretic model for intraday dispatch of grid-scale battery energy storage systems (BESSs). We assume that each BESS operator competitively manages her state-of-charge to maximize energy arbitrage revenues, driven by the endogenized electricity price that depends on the sum of the charging rates. We characterize the Nash equilibrium of the resulting finite-player linear-quadratic differential game with a shared stochastic driver, obtaining semi-explicit representations of equilibrium feedback controls and equilibrium prices both in the general heterogeneous and the simplified homogeneous BESS setting, via a system of Riccati equations. We then analyze competitive effects, including the marginal externality of additional BESS entering the market, the benefit of coordination and the corresponding market power of large operators, and supply effects from hybrid-type BESSs. We further study the asymptotic regime as the number of agents grows large. Our model provides a quantitative testbed to study the impact of decentralized BESS deployment on the grid and the resulting reduction in daily price spreads. arxiv:2605.01178 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01246v1 In this paper, we propose a single-loop stochastic gradient algorithm for solving stochastic nonconvex-concave minimax optimization with nonlinear convex coupled constraints (MCC). The proposed method, SPACO (Stochastic Penalty-based Algorithm for minimax optimization with COupled constraints), is built upon a penalty-based smooth approximation framework for MCC. This framework integrates a quadratic penalty scheme with regularization to yield a continuously differentiable approximation of the MCC problem. We provide theoretical convergence guarantees for this smoothing framework. Furthermore, we establish non-asymptotic complexity bounds and provide an asymptotic analysis characterizing the stationarity of accumulation points for the iterates generated by SPACO. Experimental results on synthetic examples and practical machine learning tasks demonstrate the effectiveness and efficiency of the proposed method. arxiv:2605.01246 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01254v1 We study hidden boundary trace regularity for two-dimensional hyperbolic equations with boundary degeneracy governed by $\mcA\vp=-\Div(A\nabla \vp)$, where $A=\diag(1,r^\al)$ and $\al\in(0,1)$. We establish well-posedness in weighted Sobolev spaces and prove an $L^2$ trace estimate for the normal derivative on the nondegenerate side $r=1$. Using truncated geometries and Carleman weights adapted to the anisotropic degeneracy, we derive a large-time observability estimate with a lower-order interior remainder. We also identify a framework-level obstruction at the critical threshold $\al=1$: the weighted Dirichlet coercivity underlying the subcritical analysis loses uniformity and exhibits a logarithmic loss on truncated domains. arxiv:2605.01254 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01344v1 This tutorial provides an overview of the generalized Lyapunov method (GLM) for analyzing input-to-state stability (ISS) of partial differential equations (PDEs). We begin by revisiting the classical Lyapunov method and the standard ISS-Lyapunov theorem, highlighting their limitations when applied to systems with complex boundary disturbances. In contrast, the GLM, based on the concept of generalized Lyapunov functionals (GLFs) that explicitly depend on the external input, offers greater flexibility and efficiency, particularly for PDEs with Dirichlet-type disturbances. The main objective of this tutorial is to demonstrate how to systematically construct GLFs to establish ISS estimates in $L^q$ spaces with any $q\in[2,\infty]$ for different PDEs. Specifically, we consider three representative classes of PDEs: (i) an $N$-dimensional nonlinear parabolic equation with mixed nonlinear boundary disturbances, (ii) a first order nonlinear hyperbolic equation with boundary disturbances, and (iii) a second order linear hyperbolic equation, i.e., a wave equation, with boundary damping and disturbances. For each case, we provide step-by-step constructions of appropriate GLFs and derive explicit ISS estimates, illustrating the general applicability of the GLM. Finally, we discuss open challenges and future directions, including the systematic construction of GLFs for broader classes of PDEs and their applications in controller design. arxiv:2605.01344 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01354v1 We prove existence and convergence of sequences generated by the proximal point method and its two variants for monotone vector fields in Hadamard spaces. Before obtaining our results, we investigate some fundamental properties of tangent spaces, resolvents, and monotone vector fields in such spaces. arxiv:2605.01354 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01377v1 We present an optimal control problem to guide the selection of morphology classes arising in organic solar cells. The study focuses on phase separation processes in polymer solvent mixtures, with particular attention to solvent evaporation as a mechanism to arrest morphology formation. We establish the existence of optimal controls and analyze the Frechet derivative of the control to state mapping. Finally, we derive the first order necessary optimality condition via the corresponding adjoint system. arxiv:2605.01377 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01467v1 Tensor completion has emerged as a powerful framework for recovering missing data in multidimensional signals by exploiting low-rank tensor structures. Among existing approaches, linear transform-based tensor nuclear norm (TNN) methods have achieved considerable success by enforcing low-rankness on transformed frontal slices. However, the low-rank structure revealed by linear transforms remains inherently limited. To better capture intrinsic correlations, nonlinear transform-based TNN (NTTNN) models have been proposed, significantly enhancing low-rank representation through composite transforms. Despite their effectiveness, existing NTTNN methods are restricted to real-valued tensors and fail to model quaternion-valued data, which are essential for preserving inter-channel dependencies in color images and videos. Extending nonlinear TNN models to the quaternion domain is challenging due to the non-commutativity of quaternion multiplication and the complexity of quaternion singular value decomposition. To address the limitations encountered in prior works, we propose a quaternion nonlinear transform-induced tensor nuclear norm (QNTTNN) via a real embedding of quaternions, enabling tractable nuclear norm definitions and efficient optimization. Building upon QNTTNN, we formulate a quaternion tensor completion model and develop a proximal alternating minimization algorithm with rigorous convergence guarantees. Extensive experiments on benchmark color video inpainting datasets validate the superior performance of the proposed method over existing approaches. arxiv:2605.01467 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01481v1 In this study, we identify a class of redundant transitivity constraints in a 0-1 integer linear programming formulation of the clique partitioning problem. The transitivity constraints in this class can be removed from the formulation without changing the optimal solution set, although each transitivity constraint defines a facet of the associated polytope. This leads to a smaller formulation that is particularly effective for instances arising from correlation clustering, where edge weights are drawn from $\{-1,1\}$. Our computational experiments show that the resulting formulation outperforms existing formulations on such instances. arxiv:2605.01481 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01493v1 Motivated by previous efforts toward mathematically analyzing the treatment of monomials in spatial branch-and-bound, we study the convex hull of the graph of a simple monomial on a nonnegative box domain in arbitrary dimension, where at most one of the variable lower bounds is positive. We give: (i) a description via linear inequalities, and (ii) a formula for the volume. arxiv:2605.01493 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01558v1 In Willems' behavioral systems theory, a dynamical system is identified with the set of all trajectories compatible with its laws of motion. For nonlinear or stochastic systems, however, the admissible trajectory set is generally nonconvex, obstructing direct optimization over the behavior. In this paper, we lift the behavioral viewpoint from trajectories to probability measures on trajectories by representing a finite-horizon dynamical system with the set of all Borel probability measures supported on its admissible trajectories. This behavioral-measure set is convex and weakly closed even for nonlinear or stochastic dynamics, because convex combinations of trajectory distributions remain dynamically admissible even when convex combinations of trajectories do not. The extreme points are precisely the Dirac masses on individual admissible trajectories, so the classical deterministic theory is embedded as the extremal skeleton of the richer measure-valued object. On this foundation we establish two core deterministic results and outline a stochastic extension based on conditional kernel consistency. First, optimal control for a prescribed initial distribution becomes a linear program over occupation measures whose dual is exactly Bellman's dynamic-programming recursion, with strong duality under compactness and continuity. Second, for controllable linear time-invariant systems under persistency of excitation, we prove a measure-level Fundamental Lemma: every probability measure on the finite-horizon behavior factors through the data Hankel matrix, reducing any optimization over trajectory distributions to an equivalent optimization over coefficient-space distributions. This is an exact data-driven reformulation requiring no model knowledge beyond a single informative trajectory; the classical Fundamental Lemma is recovered as the special case of Dirac measures. arxiv:2605.01558 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01621v1 Bilevel learning refers to machine learning problems that can be formulated as bilevel optimization models, where decisions are organized in a hierarchical structure. This paradigm has recently gained considerable attention in machine learning, as gradient-based algorithms built on the implicit function reformulation have enabled the computation of large-scale problems involving possibly millions of variables. Despite these advances, the implicit function framework relies on restrictive assumptions, notably the requirement that the lower-level problem admit a unique optimal solution for each upper-level decision. Moreover, the computation of the derivative of the lower-level optimal solution function becomes significantly more involved when the lower-level problem includes constraints. As a result, many existing bilevel learning algorithms are effective only for relatively narrow classes of problems. This paper reviews the main algorithmic ideas underlying recent progress in bilevel learning, highlighting both the key mechanisms responsible for their scalability and the limitations that arise in more general settings. We then draw connections with the broader bilevel optimization literature and discuss algorithmic techniques that may help overcome these limitations. Our aim is to bridge the gap between bilevel learning and classical bilevel optimization, thereby supporting the development of scalable methods capable of solving more general large-scale bilevel programs. arxiv:2605.01621 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01633v1 We consider an unregularized optimal control problem subject to the steady-state Navier-Stokes equations. We derive the existence of optimal solutions and prove first- and -- necessary and sufficient -- second-order optimality conditions. To approximate solutions to the optimal control problem, we consider the variational discretization scheme. We analyze convergence properties of the discretization and prove a priori error estimates for locally optimal controls that are nonsingular and which satisfy a growth condition which implies a bang-bang structure. arxiv:2605.01633 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02961v1 Most modern bridge-diffusion methods achieve finite-time transport by specifying an interpolation, Schrödinger-bridge, or stochastic-control objective and then learning the associated score or drift field with a neural network. In contrast, we identify a restricted but sufficiently broad and analytically solvable class in which the score, intermediate marginals, and protocol gradients are available in closed form without inner stochastic simulation loops and without neural networks in the optimization loop. We recast the classical linear--quadratic--Gaussian (LQG) stochastic-control structure as a transport problem of the Path Integral Diffusion (PID) type. In classical LQG control, linear dynamics, Gaussian noise, and quadratic costs lead to Riccati equations and closed-form optimal feedback. In LQ-GM-PID, we retain the linear--quadratic stochastic-control backbone, but replace terminal state regulation by a prescribed terminal probability density and allow both the initial and terminal laws to be Gaussian Mixtures (GM). Moreover, LQ-GM-PID turns bridge diffusion from a tool for terminal target matching alone into a tool for path shaping. We demonstrate this on a 2D corridor task, a 2D multi-entrance transport task, and a high-dimensional scaling study with $d=32$ and $M=16$ Gaussian-mixture terminal modes, all with sub-50\,ms analytic precompute on a laptop. We position LQ-GM-PID as an analytically solvable reference model for the state-of-the-art neural bridge-diffusion and generative-transport methods: a controlled setting in which neural approximations, score estimates, path-shaping objectives, and protocol-learning procedures can be tested against exact quantities. arxiv:2605.02961 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01816v1 We consider an optimal transport problem between laws of random probability measures: given a base cost function, we build the associated OT cost between probability measures that in turn we use to define the OT cost between probability measures over probability measures. This setting admits a finer reformulation in terms of laws of random couplings, which retain more information than ordinary couplings. One of the main contributions of the paper is the characterization of the optimal ones in terms of Kantorovich potentials. Similarly, we also introduce the strict Monge problem, whose admissible competitors are more restrictive than in the usual Monge formulation. In this setting, we will give sufficient conditions under which the value of this problem is the same as the one considered above, in the spirit of the result by A. Pratelli. Then, for $p>1$, when the underlying cost is the distance to the power $p$ in a strictly convex Banach space, we will give sufficient conditions under which the optimal random coupling is unique and induced by a solution of the strict Monge problem, resembling the Brenier theorem. arxiv:2605.01816 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01837v1 Power oversubscription is increasingly central to datacenter operation as power density grows, making it necessary to dynamically allocate limited power budgets across devices based on real-time demand. Existing approaches typically assume flat power domains, whereas in practice power distribution is hierarchical and allocation decisions must additionally respect tenant-level contractual constraints. We present nvPAX, a constrained-optimization policy that computes feasible power allocations at every control step via a three-phase hybrid QP/LP procedure. Phase I allocates power with minimum deviation from each device's power request, while respecting job priorities. Phase II fairly distributes excess power among active devices. Phase III fairly distributes any remaining power to idle devices. The rationale behind the three phases is to allow power oversubscription while maximizing datacenter utilization. On a trace-driven large-scale simulation using GPU power telemetry from a production datacenter, nvPAX runs with a mean wall-clock time of 264.69 ms per allocation interval and achieves a mean satisfaction ratio of 98.92%, outperforming static equal-share allocation and providing robustness beyond greedy proportional allocation in the presence of non-uniform hierarchical bottlenecks. arxiv:2605.01837 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01898v1 We consider constrained linear-quadratic dynamic games arising in autonomous vehicle platooning, intersection crossing and other cooperative driving scenarios. Infinite-horizon Nash equilibria are reformulated as receding-horizon affine variational inequalities with special structure. Exploiting this formulation, we design Newton-type algorithms with local quadratic convergence. The resulting methods achieve extremely fast convergence, making them well suited for real-time and embedded receding-horizon control in safety-critical traffic applications. Simulations of platooning and intersection crossing demonstrate substantial performance gains over first-order and operator-splitting approaches, hence high application potential. arxiv:2605.01898 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01903v1 We study a decentralized linear quadratic Gaussian control problem, in which a leader and a follower must steer a linear system to a target state. The target state is known only to the leader, and no explicit communication channel exists between the agents. To address the challenge posed by this asymmetric information structure, we propose an integrated communication and control (ICoCo) framework in which the control plant itself serves as a communication channel: the leader encodes the target state into its control input through an additive communication term, and the follower decodes it from the resulting state trajectory. We design an implicit coordination scheme based on joint source-channel coding ideas, and prove that the follower's estimation error decreases monotonically to zero, enabling the two agents to coordinate increasingly well and ultimately steer the system to the target state. We then formulate the design of the communication power as an optimal control problem to minimize the overall control cost. In the fully actuated leader case, we derive necessary optimality conditions and in the under-actuated case, we solve the problem numerically. Numerical results show that the proposed scheme effectively coordinates the two agents and achieves a control cost close to that of the explicit-communication lower bound. arxiv:2605.01903 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01928v1 Neural networks increasingly embed non-differentiable components (spiking neurons, quantized layers, discrete routing, blackbox simulators, etc.) where backpropagation is inapplicable and surrogate gradients introduce bias. We present PolyStep, a gradient-free optimizer that updates parameters using only forward passes. Each step evaluates the loss at structured polytope vertices in a compressed subspace, computes softmax-weighted assignments over the resulting cost matrix, and displaces particles toward low-cost vertices via barycentric projection. This update corresponds to the one-sided limit of a regularized optimal-transport problem, inheriting its geometric structure without Sinkhorn iterations. PolyStep trains genuinely non-differentiable models where existing gradient-free methods collapse to near-random accuracy. On hard-LIF spiking networks we reach 93.4% test accuracy, outperforming all gradient-free baselines by over 60~pp and closing to within 4.4~pp of a surrogate-gradient Adam ceiling. Across four additional non-differentiable architectures (int8 quantization, argmax attention, staircase activations, hard MoE routing) we lead every gradient-free competitor. On MAX-SAT scaling from 100 to 1M variables, we sustain above 92% clause satisfaction while evolution strategies drop 8--12~pp. On RL policy search, we match OpenAI-ES on classical control and retain performance under integer and binary quantization that collapses gradient-based methods. We prove convergence to conservative-stationary points at rate $O(\log T/\sqrt{T})$ on piecewise-smooth losses, upgraded to Clarke-stationary on the headline architectures and extended to the piecewise-constant regime via a hitting-time bound. These rates match the known zeroth-order query-complexity lower bounds that all forward-only methods inherit. Code is available at https://github.com/anindex/polystep. arxiv:2605.01928 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.01996v1 Multi-agent motion planning (MAMP) is an important problem for autonomous systems with multiple agents. In this work we propose a two-step method for finding optimized and kinematically feasible solutions to MAMP problems. The first step finds an initial feasible solution using state-of-the-art methods such as conflict-based search (CBS) or priority-based search (PBS), and the second step is an improvement step which improves the solution by solving a multi-phase optimal control problem (OCP) where the initial solution is used to warm-start the solver. We also propose a method for generating motion primitives in an optimized way under the constraint that the primitive durations are all multiples of the same sample time. We evaluate our proposed framework on a MAMP problem for tractor-trailer systems. We extend the safe interval path planning with interval projections (SIPP-IP) algorithm so it can handle more general cost functions and larger agents, but our results show that for the tractor-trailer system a simple lattice-based planner performs better due to less conservative collision checks. Our experiments also indicate that CBS performs better than PBS for this system as it achieves a higher success rate in environments with obstacles and had a lower average runtime, although both planners achieve solutions of similar quality after the improvement step. arxiv:2605.01996 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02019v1 In this work we consider the multi-agent motion planning (MAMP) problem with the constraint that agents arrive at their respective goals at the same time. For the special case where all agents are initially at rest we propose a two-step method for finding optimized and kinematically feasible solutions. The first step finds an initial feasible solution by applying a state-of-the-art MAMP algorithm (conflict-based search and safe interval path planning with interval projection) backward. The algorithm is complete, and we provide necessary conditions for when it is also optimal. The second step is an improvement step where a receding-horizon optimal control problem (OCP) is posed and the solution found in the first step is used to warm-start the solver. To improve scalability we propose to solve the OCP in a distributed manner using the nonlinear alternating direction method of multipliers (NADMM). We evaluate the proposed framework in numerical experiments on a car-like vehicle. The results show that the backward planning algorithm successfully finds feasible and collision-free solutions, and that the improvement step further improves the quality of the solutions. Compared to solving the OCPs in a centralized manner, using nonlinear ADMM reduces the computation time. arxiv:2605.02019 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02127v1 We introduce PF-AGD, the first parameter-free, deterministic, accelerated first-order method to achieve $O(ε^{-5/3}\log(1/ε))$ oracle complexity bound when minimizing sufficiently smooth, non-convex functions; this is the best-known bound for first-order methods on smooth non-convex objectives. Unlike existing methods possessing this rate that require a priori knowledge of smoothness constants, we use an adaptive backtracking scheme and a gradient-based restart mechanism to estimate local curvature. This yields a practical algorithm that matches best-known theoretical rates. Empirically, PF-AGD outperforms the practical variant of AGD-Until-Guilty (Carmon et al., 2017), as well as other parameter-free variants, and is a viable alternative to nonlinear conjugate gradient methods. arxiv:2605.02127 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02147v1 Sampling-based model predictive control methods like MPPI and CEM are essential for real-time control of nonlinear robotic systems, particularly where discontinuous dynamics preclude gradient-based optimization. However, these methods derive from information-theoretic objectives that are agnostic to the geometry of the control problem, leading to pathological behaviors such as mode-averaging when the cost landscape is complex. We present OT-MPC, a sampling-based algorithm that overcomes these limitations through an entropy-regularized optimal transport formulation. By computing an optimal coupling between candidate control sequences and low-cost proposals, OT-MPC refines candidates toward nearby promising samples while coordinating updates across the ensemble to maintain coverage of the solution space. We derive closed-form, gradient-free updates via the Sinkhorn algorithm, enabling real-time performance. Experiments on navigation, manipulation, and locomotion tasks demonstrate improved success rates over existing methods. arxiv:2605.02147 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02231v1 In this work, we reveal a rich combinatorial structure underlying exact minimax optimal algorithms for classical nonexpansive fixed-point problems. This viewpoint unifies all extremal optimal methods and provides a systematic and practical framework for designing new algorithms via diagrams. Specifically, we study fixed-step algorithms represented by a lower triangular matrix H, and show that the set of optimal (N-1)-step algorithms has exactly (N-1)! vertices (extremal algorithms), each of which naturally corresponds to an arc diagram, a graph that encodes its convergence proof. Using these arc diagrams, we can compose, decompose, and analyze the properties of distinct optimal vertex algorithms. Furthermore, we determine when the H-dual operation, given by taking the anti-diagonal transpose of H, preserves the optimality of a vertex algorithm, and in such cases we characterize the convergence proof of the dual algorithm. Based on this machinery, we develop new optimal algorithms with quasi-anytime guarantees; that is, they admit an increasing integer sequence such that the corresponding iterates have the optimal residual guarantees, and are additionally robust to fixed-point operators that violate nonexpansiveness. arxiv:2605.02231 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02235v1 Decentralized strategies are of interest for local decision-making over multi-vehicle networks. This paper studies mixed traffic networks of human-driven and autonomous vehicles with partial sensor measurements. The idea is to enable the group of connected autonomous vehicles (CAVs) to track the state of a group of human-driven vehicles (HDVs) via distributed consensus-based observers/estimators. Particularly, we make no assumption that the group of HDVs is locally observable in the direct neighborhood of any CAV. Then, the main contribution is to design local residual-based fault detection and isolation (FDI) at every CAV to detect possible faults/attacks in the sensor measurements. This distributed detection strategy enables every CAV to locally find possible anomalies in its taken sensor measurement with no need for a central processing unit. Two FDI logics are proposed with and without considering the history of the residuals. These FDI techniques are based on probabilistic threshold design on the residuals (in contrast to the existing deterministic threshold FDI techniques) with no assumption that the noise is of bounded support. This is more realistic in real-world multi-vehicle transportation systems. arxiv:2605.02235 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02279v1 Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing literature often presents results at a high level of abstraction, omitting the detailed coordinate-level derivations required for implementation and algorithm development. This work provides a self-contained and rigorous treatment of the foundations of Riemannian geometry, with a focus on explicit derivations tailored to Riemannian optimization. We systematically develop the key geometric structures -- including tangent and cotangent spaces, tensor calculus, metric tensors, Levi-Civita connections, curvature, and geodesics -- emphasizing step-by-step derivations in coordinates and matrix form. Building on these foundations, we derive the Riemannian gradient, Hessian, exponential map, and retraction in a form suitable for numerical computation. We further specialize these constructions to important matrix manifolds, including the Stiefel, Grassmann, and SPD (Symmetric Positive Definite) manifolds, providing explicit formulas widely used in optimization and geometric machine learning. This monograph develops a unified and implementation-oriented treatment of Riemannian geometry for optimization on manifolds. Its main contribution is the systematic organization and detailed derivation of classical geometric constructions in forms directly usable for algorithm design and numerical implementation. By connecting coordinate-level differential geometry with matrix-manifold formulas, the monograph bridges the gap between abstract theory and practical computation, and provides a reference for researchers and practitioners working in Riemannian optimization and related fields. arxiv:2605.02279 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02305v1 Minimum distance constraints (minDCs) appear in many geometric optimization problems. They pose major challenges for mixed-integer nonlinear programming (MINLP) due to their reverse-convexity. We develop new algorithms for tightening variable bounds in general MINLPs with minDCs. Because many such problems exhibit substantial symmetry, we further introduce a practical approach for handling rotation symmetries via separation of lexicographic constraints induced by Givens rotations. In a computational study, we examine the performance of the various methods and determine the scenarios in which each approach demonstrates superiority. arxiv:2605.02305 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02330v1 The rapid proliferation of omnichannel retail strategies has fundamentally transformed store replenishment operations in uncertain supply chain environments. With retail stores increasingly acting as hybrid fulfillment centers, pooled inventory allocation must absorb uncertain order realizations, constrained receiving capacities, dynamic vehicle limits, multi-tiered product priorities, and planner-controlled outbound warehouse preferences. This study frames this commercial reality as an extended constrained variant of the Multidimensional Knapsack Problem (MKP). Recognizing that exact optimization techniques such as Mixed-Integer Linear Programming (MILP) are computationally prohibitive in large-scale real-time settings, we propose a real-time scalable heuristic embedded in a computationally efficient Decision Support System (DSS) framework based on set-oriented cumulative filtering. The framework evaluates cumulative flow-through deductions, third-party logistics routing integrations, category-specific volume caps, warehouse activation filters, and user-defined warehouse priority ranks. An extensive case study within a large retail network covering 212,278 order records from June 2025 to April 2026 demonstrates the impact of the proposed methodology. Using January 2026 as the go-live cutoff, weighted ship-to-order ratio improved from 54.1% to 67.8%, weighted same-day coverage improved from 24.3% to 37.8%, and store-days with order volumes above store limits were reduced by 48.6%. These findings indicate that the proposed real-time scalable heuristic and computationally efficient DSS framework provide practical, uncertainty-aware allocation support for volatile retail supply chains. arxiv:2605.02330 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02355v1 Regenerating braking energy is one major pathway to make rail traffic energy-efficient. It is therefore desirable to design timetables that exploit this feature. However, timetables that allow to regenerate energy are often bad for the passengers. We hence formulate and analyze a bicriteria optimization problem (PESP-Passenger-Energy) to find periodic railway timetables that maximize the regenerated energy in terms of the brake-traction overlap time and minimize the travel time of the passengers. Our model extends the Periodic Event Scheduling Problem (PESP) and offers a rich combinatorial theory. We investigate its computational complexity on one-station networks, building on matchings and Hamiltonian paths. Besides showing its NP-hardness even for a single objective, we identify several polynomial-time solvable special cases. Finally, we provide two case studies, underlining the practicability of our model, and analyzing the Pareto front. arxiv:2605.02355 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02456v1 We study the maximum $k$-colorable subgraph (M$k$CS) problem, which consists in finding a largest $k$-colorable induced subgraph in a given graph. We consider a Semidefinite Programming (SDP) relaxation for the M$k$CS problem and regard its resulting upper bound as a graph parameter. We present several properties of this graph parameter, from which we obtain that the M$k$CS problem is solvable in polynomial time for $k$-perfect graphs. We further derive two novel families of valid inequalities to strengthen the SDP relaxation. The first family reduces to a family of inequalities for the Boolean quadric polytope when $k = 1$, and the second family generalizes the family of rank inequalities for binary linear programming formulations of the stable set problem. We efficiently solve the strengthened SDP relaxation using a cutting-plane algorithm that is based on the Alternating Direction Method of Multipliers (ADMM). Extensive computational experiments show that the obtained upper bounds outperform the best upper bounds from the literature. To complement our SDP-based upper bounds, we propose an integer ADMM variant that uses an exact Binary Semidefinite Programming (BSDP) formulation of the M$k$CS problem to produce high-quality feasible solutions. To the best of our knowledge, this is the first application of the ADMM to compute integer solutions to a BSDP problem. arxiv:2605.02456 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02480v1 We study the observability of the Schrödinger equation on the $d$-dimensional torus $\mathbb T^d$, $d \geq 1$, from an open subset $ω\subset \mathbb T^d$. Our first main result establishes a quantitative observability estimate for the free Schrödinger equation in the regime of small times $T$ and for small observation sets of the form $ω= \prod_{j=1}^{d}(a_j,b_j)$. Our second main result shows that observability holds for the Schrödinger equation with a merely bounded potential $V \in L^{\infty}(\mathbb T^d)$, in any dimension $d \geq 1$, for every time $T>0$ and every nonempty open subset $ω$. This resolves a well-known conjecture in the field. A central ingredient in the proof is a cluster decomposition method combined with an induction scheme introduced by Bourgain and further developed by Burq and Zhu. arxiv:2605.02480 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02497v1 We obtain an explicit solution for the static Kullback--Leibler (KL) unbalanced optimal transport problem between finite non-degenerate Gaussian measures with quadratic cost, two independent positive marginal relaxation parameters, and no entropy penalty on the coupling. The minimizer is a scaled Wasserstein coupling between two adjusted Gaussian marginals and is supported on an affine graph; in entropic Gaussian unbalanced transport, by contrast, the optimal plan is non-degenerate on the product space. The covariance map is the unique positive definite solution of a Riccati equation and admits a principal-square-root representation. Compared with the known equal-penalty Gaussian Hellinger--Kantorovich endpoint, the result treats the asymmetric two-sided Kullback--Leibler relaxation and gives the modified marginals, joint minimizer, value, and a direct quadratic KL-dual certificate. The large-relaxation limit recovers the Gaussian Wasserstein cost for equal masses. arxiv:2605.02497 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02582v1 We study optimal decision policies for integer linear programs with a fixed feasible set and varying cost vectors, represented as linear decision trees. Once synthesized for a given feasible set, they return an optimal solution for any queried cost vector through a sequence of linear tests. We show that there exists a policy performing this operation in a polynomial number of arithmetic operations in the worst case. Along with this theoretical guarantee, we develop a practical construction framework to synthesize policies within a specific subclass of linear decision trees. Our computational experiments show that, although policy synthesis can be time-intensive, it allows retrieving optimal solutions orders of magnitude faster than classical and specialized solution methods on repeated queries. Overall, this paradigm provides a new perspective on the complexity of integer linear programs and offers an offline--online approach for solving them. arxiv:2605.02582 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02653v1 We study an explicit mirror-descent method for finite-horizon deterministic optimal control problems. The method is motivated by Pontryagin's maximum principle: at each iteration, one solves the state and adjoint equations and updates the control by maximizing a first-order approximation of the regularized Hamiltonian penalized by a Bregman divergence. In the Euclidean case, the update reduces to a projected gradient step in the control variable. Under global smoothness assumptions and uniform convexity of the mirror map, we prove a relative smoothness estimate for the cost functional and derive an energy dissipation inequality for sufficiently small step sizes. Under an additional concavity assumption on the unregularized Hamiltonian and convexity of the terminal cost, we establish relative convexity of the regularized objective. These estimates yield an $O(1/n)$ convergence rate in the unregularized convex case and a geometric rate when the control regularization parameter is positive. Numerical examples illustrate the behavior of the method in linear-quadratic, degenerate convex, and nonlinear high-dimensional settings. arxiv:2605.02653 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02701v1 We propose a robust gradient estimator based on per-sample gradient clipping and analyze its properties both theoretically and empirically. We show that the resulting method, per-sample clipped SGD (PS-Clip-SGD), achieves optimal in-expectation convergence rates for non-convex optimization problems under heavy-tailed gradient noise. Moreover, we establish high-probability convergence guarantees that match the in-expectation rates up to polylogarithmic factors in the failure probability. We complement our theoretical results with multiple numerical experiments. In particular, we demonstrate that PS-Clip-SGD outperforms both vanilla SGD with momentum and standard gradient clipping when training AlexNet on the CIFAR-100 dataset, even after accounting for the additional computational time caused by per-sample clipping. We also empirically show that, in the presence of gradient accumulation, applying clipping at the mini-batch level can improve training performance while incurring virtually no additional computational cost. This finding is particularly interesting, as it contradicts the common practice of applying clipping only after all accumulation steps have been completed. arxiv:2605.02701 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02754v1 Identifiability means that iterates generated by optimization algorithms are eventually confined to an identifiable set. This property is computationally useful because minimizing a nonsmooth function near a critical point reduces to minimizing its smooth restriction on the corresponding identifiable manifold. Motivated by this reduction, we study the Error Bound (EB) property from both ambient and manifold viewpoints. Under mild assumptions in Euclidean space, we prove that local EB on $(\mathbb{R}^n,d)$ is equivalent to local EB on an identifiable manifold $(\mathcal{M},d)$. We establish this result from two complementary perspectives: a metric analysis based on slope and linear growth away from $\mathcal{M}$, and a geometric analysis based on subdifferentials, partial smoothness, and $\mathcal{VU}$-theory. As an application, we recover the EB equivalence for $\ell_1$-regularized optimization in the literature. arxiv:2605.02754 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02783v1 In this paper, we focus on two types of degenerate partial differential equations: a degenerate elliptic equation and a degenerate parabolic equation. Significantly, both categories are characterized by the same principal operator. To obtain solutions for these equations, we introduce a novel approximation approach, termed the shape design approximation. As a practical application of this method, we derive a Carleman estimate for the backward degenerate parabolic equation. This estimate plays a pivotal role in establishing the null controllability of the degenerate parabolic equation. A notable advantage of employing the shape design approximation in deriving the Carleman estimate is that it enables us to bypass the requirement for second order derivatives in the degenerate equation. Usually, this has been a significant obstacle in the derivation of Carleman estimates for degenerate parabolic equations. arxiv:2605.02783 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02791v1 A number of important modern applications in optimal control can be formulated as open loop control problems in which the underlying dynamical systems are subject to random inputs. These so-called ensemble control problems require the corresponding optimal control to be deterministic, as it must be computed before the realization of uncertainty and the passage of time. Practical applications of ensemble control include quantum control and the training of Neural ODEs. However, the standard approach to ensemble control treats the uncertainty in the objective function via the expectation, which provides optimal controls that only work well on average while ignoring critical outlier phenomena. This study provides a comprehensive mathematical treatment of risk-averse ensemble control. Within this setting, we adopt a control-affine structure that ensures the lower semi-continuity needed for proving the existence of optimal solutions. The central analytical contribution of this paper is a rigorous characterization of the control-to-state mapping in which we establish weak-to-strong continuity, continuous Fréchet differentiability, and weak-to-strong continuity of the derivative operator. Furthermore, this regularity yields primal and dual first-order optimality conditions characterized by an adjoint state of bounded variation, and it fulfills the functional prerequisites required for the convergence of infinite dimensional optimization algorithms. We conclude by validating these theoretical developments through a numerical experiment in quantum control. arxiv:2605.02791 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02797v1 In this paper, we establish a quantitative weak unique continuation theorem on an annular domain for a backward degenerate parabolic equation with a degenerate interior point. Our methodology hinges on approximating the solution of the degenerate parabolic equation through solutions of non-degenerate parabolic counterparts. Subsequently, we establish Carleman estimates for the non-degenerate parabolic equation across two separate domains. By virtue of these estimates, we deduce a quantitative weak unique continuation property for the degenerate parabolic equation, thereby substantiating the weak unique continuation result for the original degenerate parabolic equation. arxiv:2605.02797 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02806v1 Day-to-day traffic dynamics are widely used to model flow evolution due to travelers' learning and adjustment behavior, yet empirical analysis of these models often relies on descriptive calibration with limited inferential content. This paper develops a statistical inference framework for day-to-day route choice dynamics based on a stochastic individual-level adjustment model. The framework enables uncertainty quantification and formal inference for behavioral parameters from trajectory data. We establish identifiability and consistency under mild conditions, and extend the framework to accommodate demand variation, user heterogeneity through a hierarchical structure, and anonymized observability caused by privacy constraints on trajectory data. Simulation studies demonstrate good finite-sample performance, calibrated uncertainty, and robustness to model misspecification. Empirical analyses of controlled laboratory experiments and real-world trajectory data from Ann Arbor, Michigan, show that the framework can generate novel behavioral insights across settings: it reveals the inadequacy of a purely inter-day learning model once en-route information is introduced, recovers systematic behavioral differences across participant types, and uncovers meaningful day-to-day learning together with substantial demand variation in real-world commuting behavior. arxiv:2605.02806 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02830v1 In this study, we study the null controllability of a multi-dimensional degenerate parabolic equation characterized by a degenerate interior point. The control domain, which is an arbitrary inner region, does not encompass the degenerate point. To tackle this problem, we adopt a new approximation methodology. Specifically, we approximate the degenerate partial differential equations (PDEs) with a series of uniformly elliptic PDEs, notwithstanding their limited regularity. We then derive the Carleman estimate for these approximate uniformly parabolic equations and establish the observability inequality, which ultimately paves the way for demonstrating the null controllability of the system. arxiv:2605.02830 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.02838v2 Retraction-free approaches offer attractive low-cost alternatives to Riemannian methods on the Stiefel manifold, but they are often first-order, which may limit the efficiency under high-accuracy requirements. To this end, we propose a second-order method landing on the Stiefel manifold without invoking retractions, which is proved to enjoy local quadratic (or superlinear for its inexact variant) convergence. The update consists of the sum of (i) a component tangent to the level set of the constraint-defining function that aims to reduce the objective and (ii) a component normal to the same level set that reduces the infeasibility. Specifically, we construct the normal component via Newton$\unicode{x2013}$Schulz, a fixed-point iteration for orthogonalization. Moreover, we establish a geometric connection between the Newton$\unicode{x2013}$Schulz iteration and Stiefel manifolds, in which Newton$\unicode{x2013}$Schulz moves along the normal space. For the tangent component, we formulate a modified Newton equation that incorporates Newton$\unicode{x2013}$Schulz. Numerical experiments on the orthogonal Procrustes problem, principal component analysis, and real-data independent component analysis illustrate that the proposed method performs better than the existing methods. arxiv:2605.02838 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03072v1 The design of tactical wireless networks reflects a complex interplay among structural constraints, technological choices, and underlying modeling assumptions. Although optimization-based approaches have been widely explored, the impact of configuration parameters on network topology quality and overall performance is still not fully understood. This paper presents a comprehensive sensitivity analysis of tactical wireless network design under realistic operational constraints. It systematically investigates three categories of parameters: (i) structural topology rules, including master hub selection; (ii) technological factors such as antenna beamwidth; and (iii) modeling parameters embedded in the objective formulation. Optimized topologies are produced using a Tabu Search metaheuristic, and statistical analyses based on the Friedman and Wilcoxon tests are performed to assess the significance of observed variations across different network sizes. The findings reveal scale-dependent technological transitions and threshold effects in structural constraints. The analysis differentiates parameters that fundamentally reshape network topology from those that primarily influence performance magnitude. Together, these insights provide practical guidance for parameter tuning and topology configuration in mission-critical tactical wireless deployments. arxiv:2605.03072 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03263v1 We propose MultiLRSGA, an $h$-player extension of LRSGA for the computation of stable Nash equilibria in differentiable games. The method originates from the decomposition of the game Jacobian into symmetric and antisymmetric components, which motivates symplectic corrections designed to attenuate the rotational part of the dynamics. In the two-player setting, LRSGA replaces mixed second-order blocks with low-rank secant approximations. The passage to the multi-player case, however, is not a mere blockwise reformulation: the antisymmetric correction is no longer determined by a single pair of cross-interactions, but by a block antisymmetric operator collecting all pairwise couplings among the players. On this basis, we formulate MultiLRSGA by constructing, for each player, a low-rank approximation of the Jacobian of the partial gradient and extracting from it the blocks required to define an approximate antisymmetric correction. Under standard local assumptions around a stable Nash equilibrium, we prove local linear convergence of the method. The key technical ingredient is a lemma controlling the distance between the exact antisymmetric correction and its secant approximation in the $h$-player setting, thereby extending to the multi-player framework the convergence mechanism previously available for LRSGA. The proposed formulation preserves the computational advantages of low-rank symplectic corrections and is naturally suited to numerical validation on differentiable games with explicit payoffs and more than two agents. arxiv:2605.03263 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03357v1 Mean Field Games (MFGs) provide a powerful framework for modeling the collective behavior of large populations of interacting agents. In this paper, we address the problem of Imitation Learning (IL) in MFGs subject to common noise, where the population distribution evolves stochastically. This stochasticity compels agents to adopt population-aware policies to respond to aggregate shocks. We formulate two distinct learning objectives: recovering a Nash equilibrium and maximizing performance against an expert population. We investigate two imitation proxies: Behavioral Cloning (BC) and Adversarial (ADV) divergence. We then establish finite-sample error bounds showing that minimizing these proxies effectively controls both the policy's exploitability and its performance gap relative to the expert. Furthermore, we propose a numerical framework using generalized Fictitious Play and Deep Learning to compute expert population-aware policies. Through experiments on three environments we demonstrate that standard population-unaware policies fail to capture the equilibrium dynamics. Our results highlight that learning population-aware policies is crucial to avoid being misled by the randomness inherent in common noise. arxiv:2605.03357 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03400v1 We propose a novel stochastic approximation algorithm, termed PMQSopt, for solving weakly convex stochastic optimization problems involving expectation-valued functions. The algorithm is constructed by integrating the proximal method of multipliers with quadratic approximations of the original stochastic problem. We analyze the sample complexity of PMQSopt in terms of the total number of stochastic gradient evaluations required. The convergence of the algorithm is characterized by three metrics associated with the $ε$-KKT conditions: the average squared norm of the gradient of the Moreau envelope of the Lagrangian, the average constraint violation, and the average complementarity violation. For each of these metrics, we establish an expected convergence rate of $\mathcal{O}(T^{-1/4})$ after $T$ iterations. Furthermore, we show that with probability at least $1-1/T^{2/3}$, the gradient of the Lagrangian satisfies an $\mathcal{O}(T^{-1/8})$ bound; with probability at least $1-2/T^{2/3}$, the constraint violation achieves an $\mathcal{O}(T^{-1/4})$ bound; and with probability at least $1-3/T^{2/3}$, the complementarity violation attains an $\mathcal{O}(T^{-1/4})$ bound. All results are established under two mild conditions: (i) weak convexity of all problem functions, and (ii) the existence of a strictly feasible point. The proposed PMQSopt algorithm is a sequentially strongly convex programming method that is readily implementable. Numerical experiments illustrate its practical performance. arxiv:2605.03400 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03415v1 This paper proposes QPALM, a proximal augmented Lagrangian method based on quadratic approximations, for solving nonlinear programming problems with weakly convex objective and constraint functions. The algorithm is constructed by incorporating quadratic approximations of both the objective and constraint functions into a proximal Lagrangian framework. We establish its non-asymptotic convergence rate in terms of the total number of subproblems solved. The convergence of QPALM is characterized by three metrics associated with the $\varepsilon$-KKT conditions: the squared norm of the gradient of the Moreau envelope of the Lagrangian, the average constraint violation, and the average complementarity violation. All three metrics are shown to converge at a rate of $O(T^{-1/3})$ after $T$ iterations. Preliminary numerical results demonstrate the practical efficiency of the proposed method. These results are established under two mild conditions: (i) weak convexity of all problem functions, and (ii) the existence of a strictly feasible point. The proposed QPALM is a sequentially strongly convex programming method that is readily implementable. arxiv:2605.03415 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03436v1 Allocating scarce, indivisible resources to diverse groups under uncertainty is a central challenge in operations research, where efficiency-focused methods often underserve marginalized populations. We study the Fair Online Resource Allocation with Indivisible Units (FORA-IU) problem, in which an unpredictable sequence of demands must be served from a strictly fixed inventory, and ask what fairness guarantees are achievable under different distributional and structural assumptions. We adopt a fairness criterion based on the expected filling ratio (FE-FR-beta), which balances each group's expected allocation against its expected demand and priority weight. We design online policies that calibrate acceptance probabilities to the remaining budget, analyze both arbitrary time-varying and stationary arrivals, introduce the Random Cyclic Blocks (RCB) algorithm tailored to the stationary case, and study the effect of restricting policies to all-or-nothing allocations. For arbitrary time-varying arrivals, our policy achieves the optimal universal fairness guarantee of 1/(1+R_beta), where R_beta denotes the priority-weighted system load. For time-invariant arrivals, RCB achieves the exact finite-horizon guarantee [1-(1-R_beta/T)^T]/R_beta, which is at least (1-e^{-R_beta})/R_beta and is also tight. We further show that all-or-nothing allocation policies cannot match these guarantees. These findings demonstrate that distributional stationarity strictly improves the fairness frontier, and that partial fulfillment is a necessary condition for attaining optimal fairness in online indivisible resource allocation. arxiv:2605.03436 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03484v1 We provide sufficient conditions for quantitative convergence of the iterates of proximal splitting algorithms for minimizing a sum of functions on a metric space. The theory does not assume that the functions have common minima, nor does it require vanishing proximal parameters or step sizes. Our results are stated for general $p$-uniformly convex spaces with curvature bounded above, and a corollary specializes the main theorem to Hadamard spaces, where many assumptions for the more general setting can be dropped. The theory is demonstrated with computation of Fréchet means in the space of SPD matrices with the affine invariant metric (a Hadamard space) and the sphere with the usual geodesic metric (a CAT($κ$) metric space). arxiv:2605.03484 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03520v1 We propose a neural parameterization of convex sets by learning sublinear (positively homogeneous and convex) functions. Our networks implicitly represent both the support and gauge functions of a convex body. We prove a universal approximation theorem for convex sets under this parametrization. Empirically, we demonstrate the method on shape optimization and inverse design tasks, achieving accurate reconstruction of target shapes. arxiv:2605.03520 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03711v1 Spline functions are smooth piecewise polynomials widely used for interpolation and smoothing, and nonnegative spline smoothing is also studied for nonnegative data. Previous research used sufficient conditions for the nonnegativity of spline functions because necessary and sufficient conditions for the nonnegativity are infinitely many linear inequalities, which are difficult to handle in optimization algorithms. This conventional method quickly computes a nonnegative spline function via quadratic programming (QP), but the optimal solution may be slightly degraded by using the sufficient condition. In this paper, we express 1D nonnegative spline smoothing as a convex semi-infinite programming (CSIP) problem that directly deals with infinite inequality constraints. As optimization algorithms for general SIP problems, local-reduction-based sequential quadratic programming (LRSQP) methods are used, but their convergence performance deteriorates for certain problems due to multiple approximations during updates. To quickly solve the CSIP problem, we propose a cutting-plane (CP) method. In the proposed method, after giving an initial solution by the standard spline smoothing, we find the minimizer of each polynomial piece by using the closed-form solution for a low-degree polynomial or a numerical solution for a high-degree polynomial. If the minimum value is negative, then such minimizer is added into the constraint of the problem to guarantee the nonnegativity. This constrained problem is quickly solved via QP, and we find the minimizer of each polynomial piece again. We repeat these procedures until there are no negative minimum values. The proposed method guarantees convergence to the original CSIP solution, and its effectiveness is demonstrated in numerical experiments by comparison to the conventional methods, QP under the sufficient condition and CSIP using the MATLAB LRSQP algorithm. arxiv:2605.03711 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03726v1 We study the problem of global exponential stabilization of a force- and torque-controlled unicycle model in discrete time. To this end, we extend a recently introduced approach to model predictive control (MPC) in which a flexible number of inputs is implemented in every iteration. We present the first flexible-step MPC protocol with state-dependent weights for average descent. Notably, the proposed method relies neither on a suitable design of running or terminal cost functions nor on a suitable choice of terminal constraints. Instead, stability is guaranteed through a generalized discrete-time control Lyapunov function. We establish a new theoretical framework for global exponential stabilization of general nonlinear discrete-time control systems by flexible-step MPC. The obtained results go beyond the unicycle example. However, given the importance of the unicycle dynamics, we make that a focal point of our work. For the particular case of the dynamic (second-order) unicycle model, we show that global exponential stability cannot be attained in the classical sense, but in a slightly weaker sense. The proposed flexible-step MPC method is shown to induce the best possible notion of global exponential stability for this model. We provide explicit rules for the choice of parameters, which guarantee feasibility and global exponential stability. Our numerical simulations show that the discrete MPC method also works very well in applications to a continuous-time torque-actuated unicycle. arxiv:2605.03726 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03753v1 We address day-ahead transmission topology planning and congestion management as a sequential, multi-objective optimization problem and develop two complementary algorithms for it: an exact enumeration method and a tailored evolutionary heuristic. The problem is formulated with four operational objectives reflecting real TSO decision criteria: worst-case line loading under $N-1$ security, topological depth, number of switching actions, and time spent in non-reference topologies, over a 24-hour horizon. We introduce the block algorithm, an exact method that exploits the temporal block structure of feasible strategies to enumerate the complete Pareto front; for fixed operational bounds on depth and switch count, its evaluation count grows polynomially with the planning horizon. We complement it with a multi-objective evolutionary algorithm based on NSGA-III, with structure-guided initialization and problem-specific variation operators tailored to the topology-planning structure. Using real operational data from the Dutch high-voltage grid operated by TenneT TSO, we show that the block algorithm computes the full Pareto front for a highly congested day in under three minutes, and that the evolutionary algorithm converges toward but does not recover the exact front. The block algorithm thus provides both a practical decision-support tool and a ground-truth benchmark for future heuristic and learning-based methods on this problem class. arxiv:2605.03753 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03760v1 In this work, we investigate whether machine learning can be leveraged to identify promising states in dynamic programming algorithms, focusing on Elementary Resource Constrained Shortest Path Problems (ERCSPP). More in detail, we solved 41 single resource instances from SPPRCLIB using iterative relaxation techniques through the PathWyse library, systematically collecting all generated states (i.e. labels). We designed ad-hoc features computable in constant time and constructed two datasets: one containing all generated labels (G) and another with only those inserted into data pools (I), totaling several hundred million labels. Machine learning tools are then employed to explore these datasets, revealing significant patterns between successive relaxations. Leveraging these insights, we propose a normalization approach and apply supervised learning techniques to distinguish dominating states, both within subsequent relaxations of the same problem and in previously unseen instances. Our results demonstrate the effectiveness of this approach on Dataset G, while for Dataset I, performance varies, showing strong results within the same instance but declining for unseen ones. Overall, these findings open new perspectives for the development of data-driven dynamic programming algorithms. arxiv:2605.03760 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03772v1 We study the induced matrix norm $\|\bA\|_{q \to r}$, whose exact value has been known only in a few classical cases. Determining this norm has long been regarded as difficult due to the highly non-convex nature of its variational definition. Existing works offer numerical estimates or analytic bounds but no exact formula. In this paper we present a purely analytic framework that determines $\|\bA\|_{q \to r}$ exactly for all $q, r \ge 1$ for several classes of important matrices. For these matrices, using a direct connection between the induced norms and Grothendieck problems, our results also simultaneously provide exact values for the later. arxiv:2605.03772 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03773v1 The computation of quantum entanglement can be formulated as a high-dimensional nonconvex optimization problem with orthogonality constraints. In this work, we propose structure-preserving consensus-based optimization (CBO) methods for entanglement computation, with one approach based on a Hermitian formulation and the other evolving directly on the unitary manifold. To handle the variable dimension of the feasible set, we introduce a cross-dimensional interaction mechanism allowing exchange of information between particles of different sizes. Numerical experiments demonstrate that the proposed methods achieve accurate approximations. arxiv:2605.03773 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03785v1 We study the cyclic inventory routing problem that involves joint decisions on vehicle routing and inventory replenishment on an infinite, cyclic horizon. It considers a single warehouse and a set of geographically dispersed retailers. We model retailer demand as random variables with uncertain distributions belonging to a moment-based ambiguity set. We develop a distributionally robust optimization formulation that minimizes the worst-case expected cost over the ambiguity set, while ensuring service reliability through a distributionally robust chance constraint. Our main results are that we prove that the worst-case expected inventory cost is attained under a multi-point distribution, which can be identified a posteriori via linear programming, and that the distributionally robust chance constraint can be reformulated into near-equivalent deterministic forms. This yields a deterministic reformulation of the original problem. To solve it, we design a nested branch-and-price framework, in which the first level partitions retailers into clusters, and the second level concerns routing and replenishment decisions within each cluster. Computational experiments on both synthetic instances and real-world data from SAIC Volkswagen Automobile Co., Ltd. demonstrate the effectiveness and efficiency of the proposed approach. arxiv:2605.03785 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03797v1 We develop a functional extension of an extremal principle by Schneider (Monatsh. Math., 1967) by introducing generalized outer linearizations of convex functions. Given a coercive convex function on $\mathbb{R}^n$, a generalized outer linearization is defined as a convex minorant represented by a general but function-dependent set of slopes, thereby extending classical outer representations of convex bodies by supporting halfspaces. This representation converts geometric outer approximations by supporting halfspaces into functional approximations by supporting affine functions, and replaces outer normal data by a dual sampling problem in the domain of the Legendre--Fenchel transform. On a standard class of coercive convex functions, we derive a general extremal principle, showing that the rotational epi-symmetrization maximizes best approximations under outer linearizations of any monotone, concave functional that is upper semicontinuous with respect to epi-convergence. A central feature of the analysis is that it is carried out in the natural class of coercive, but not necessarily super-coercive, convex functions. Working in this setting introduces intricate topological and variational difficulties, which are addressed using refined duality and epi-convergence arguments. As an application of our main results, we derive a functional version of Urysohn's inequality, as well as an analytic extension of a classical covering result of Firey and Groemer (J. London Math. Soc., 1964). Finally, we prove an extremal inequality related to the piecewise affine approximation of convex functions. arxiv:2605.03797 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03815v1 We consider the problem of minimizing a proper, lower semicontinuous, geodesically convex function on a Hadamard manifold. Building on ball-proximal (broximal) ideas in the Euclidean setting, viewed as an abstract proximal-type algorithm, we propose and analyze a Riemannian ball-proximal point method (RB-PPM) whose basic step consists of minimizing the objective function over a metric ball centred at the current iterate. We first introduce the Riemannian broximal map, prove existence and uniqueness of broximal points on Hadamard manifolds, and derive a KKT-type characterization involving a scalar parameter and the Riemannian subdifferential. We then show that RB-PPM enjoys a strict decrease of the squared distance to the solution set whenever the current ball does not contain a minimizer. This leads to quasi-Fejér monotonicity, finite termination for constant radii, and a product-form linear decay of the objective values up to the hitting time of the solution set. We also obtain nonasymptotic complexity bounds for the norms of suitable subgradients and for the function values, including a linear rate in the number of iterations under constant radii. Finally, we establish an asymptotic dichotomy, if the sum of the radii diverges, then the objective values converge to the optimal value, and, when the solution set is nonempty, the entire sequence of iterates converges to a minimizer. The resulting scheme provides a geometry-aware, ball-based analog of classical Riemannian proximal point methods. arxiv:2605.03815 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03869v1 We investigate the effectiveness of adaptive zeroth-order (ZO) optimization for memory-constrained fine-tuning of large language models (LLMs). Contrary to prior claims, we show that adaptive ZO methods such as ZO-Adam offer no convergence advantage over well-tuned ZO-SGD, while incurring significant memory overhead. Our analysis reveals that in high dimensions, ZO gradients lack coordinate-wise heterogeneity, rendering adaptive mechanisms memory inefficient. Leveraging this insight, we propose MEAZO, a memory-efficient adaptive ZO optimizer that tracks only a single scalar for global step size adaptation. We support our method with theoretical convergence guarantees under standard assumptions. Experiments across multiple LLM families and tasks demonstrate that MEAZO matches ZO-Adam's performance with the memory footprint of ZO-SGD. Additional experiments on synthetic quadratic problems and LLM fine-tuning further demonstrate MEAZO's enhanced robustness to step size choices, particularly in grouped or block-structured optimization settings. arxiv:2605.03869 Wed, 06 May 2026 08:25:17 +0000 http://arxiv.org/abs/2605.03959v1 The constrained generalized maximum-entropy sampling problem (CGMESP) is to select an order-s principal submatrix from an order-n covariance matrix, subject to some linear side constraints, so as to maximize the product of its t greatest eigenvalues, 0 < t <= s <n. GMESP refers to the version with no side constraints. Introduced more than 25 years ago, CGMESP is a natural generalization of two fundamental problems in statistical design theory: (i) constrained maximum-entropy sampling problem (CMESP); (ii) binary D-optimality (D-Opt). In the general case, it can be motivated by a selection problem in the context of principal component analysis (PCA). We present novel non-convex extended variable formulations for CGMESP. Using these formulations as points of departure, we present, first non-convex and then convex, continuous relaxations for CGMESP. We demonstrate many relations between different upper bounds for CGMESP, including upper bounds from the literature and our new upper bounds. We investigate the behavior of our relaxations related to the constraints linking the natural variables with the extended variables. We propose and investigate a generalized scaling technique for bound improvement. In the context of branch-and-bound, we determine the better of two natural branching techniques for fixing variables to zero. Finally, we present numerical experiments illustrating the value of our methods. arxiv:2605.03959 Wed, 06 May 2026 08:25:17 +0000