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math.OC

math.OC: explore 248 source-linked works published from 2017 to 2026, with original documents and citations.

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Includes records with this source-supplied label or an explicit phrase match in their metadata. Matches indicate a mention, not proof that a paper uses a method or tests a material. Source versions are consolidated by DOI.

Sources: arxiv. Collection updated 2026-09-16. Counts describe this index, not the complete source archives.

arXiv2026

Projected Riemannian Gradient Descent for the Bures-Wasserstein Barycenter: Dimension-Independent Linear Convergence at Unit Step Size

The computation of the Bures-Wasserstein (BW) barycenter of an ensemble of positive definite matrices arises throughout machine learning, optimal transport, and quantum information. Riemannian gradient descent (RGD) at unit step size -- the fixed-point iteration used in practice -- converges rapidly, yet existing analyses present a dichotomy: unit-step guarantees carry worst-case exponential dependence on the dimension, while dimension-independent guarantees require small step sizes that forfeit the empirical speed. We resolve this dichotomy, not by improving the guarantees for unit-step RGD, but by proposing a Projected RGD algorithm that achieves dimension-independent linear convergence at unit step size. The achieved rate, $(1 - κ^{-3/2})$, where $κ$ is the condition number of the ensemble, also polynomially improves on the best small-step guarantee ($κ^{3/2}$ versus $κ^{5/2}$ iteration complexity). The crux is a novel Projection Lemma: clipping the eigenvalues of a positive matrix to an interval $[α, β]$ is the closed-form, non-expansive (1-Lipschitz) BW-metric projection onto the set $\{S : αI \leq S \leq βI\}$ -- a statement which, unlike its known one-sided counterpart, does not follow from convexity. The projection is moreover free: it reuses an eigendecomposition the next iteration must perform in any case, so the projected and unprojected iterations cost the same per step. The same analysis covers the invariant matrix projection problem of Brahmachari et al. (2025), whose fixed-point algorithm we identify as unit-step RGD on a totally geodesic submanifold, thereby extending the dimension-independent guarantee to that setting verbatim.

cs.LG↗

arXiv2026

Adapting to Evolving Requirements: Agentic AI for Retail Supply Chain Operations

Retail supply chain operations rely on coupled decision modules that must adapt as requirements evolve. LLMs offer a natural-language interface for this task, but existing methods primarily focus on individual optimization models. Extending them to heterogeneous decision pipelines is challenging because a requirement may admit multiple intervention paths with different downstream effects. We formulate requirement-driven adaptation as the joint selection of an intervention route and an admissible module-level change, and propose a graph-constrained agentic framework in which domain agents expose admissible reformulation interfaces and a central processor searches over bounded intervention paths. Candidates are validated and compared using downstream KPIs. In collaboration with a large retail partner, we evaluate 100 warehouse requirements elicited from practitioner interviews, with GPT, Qwen, and DeepSeek as base LLMs. Relative to direct LLM reformulation, our framework improves correctness and end-to-end success across all three models, raising end-to-end success from 72--76% to 79--83%.

cs.AI↗

arXiv2026

Transversality Conditions for Boundary Constraints Defined by Differential Equations

What are the transversality conditions for an optimal control problem when the boundary conditions are defined by differential equations? This seemingly bizarre question is motivated by trajectory optimization problems in the $N$-body system. The question, however, is more fundamental and goes beyond problems in astrodynamics to nonintegrable dynamical systems in general. The main contribution of this paper is the development of generic initial- and final-time transversality conditions for optimal control problems whose boundary conditions are defined in terms of differential equations with side conditions. The mathematical definition of differential boundary conditions are part of the foundations developed in this paper. To support the new fundamentals, the concept of coordinated/uncoordinated clock times and weak adjoint covectors are introduced. In the case of uncoordinated clock times, the new transversality conditions reveal that there exists a special situation where a weak adjoint covector is orthogonal to the vector field of the boundary differential equation. This condition is sharply different from the classical statement of orthogonality with respect to the endpoint manifold. The theorems developed in this paper are generic. An application of the theorems to several cases in the three-body problem are described in separate papers.

math.OC↗

arXiv2026

Minimizing the makespan in job shop scheduling under conflict graph constraints

We study the job shop scheduling problem with a conflict graph, in which adjacent jobs cannot be processed simultaneously on different machines, with the objective of minimizing the makespan. We establish a polynomial equivalence with a resource-constrained job shop problem having unit-capacity resources and binary resource requirements The problem is strongly NP-hard even on two machines with unit-time operations. For this setting, we derive an optimal-makespan formula and a linear-time algorithm when the conflict graph is a cluster graph, given the clique partition. We also develop a polynomial-time $4/3$-approximation algorithm for arbitrary conflict graphs on two machines when each job has exactly two unit-time operations. For the general problem, we propose four mixed-integer linear formulations, lower bounds, and a genetic algorithm with active, non-delay, and hybrid decoding procedures. Computational experiments on instances derived from the Lawrence and Taillard benchmarks and on randomly generated generalized job shop instances evaluate the proposed methods and show that the genetic algorithm generally improves on constructive heuristics.

cs.DM↗

arXiv2026

Approximating CDTW Distance of Piecewise Algebraic Curves

Curves as input data naturally arise in a variety of fields including finance, seismology, medicine, spatio-temporal data mining, malicious activity detection, and more. A common way to analyze these data sets is to do similarity matching or clustering. The most common metrics used for measuring similarity of curves are Dynamic Time Warping (DTW) and Fréchet distance. These metrics are sensitive to sampling rate and outliers respectively, and do not yield robust outcomes. Continuous Dynamic Time Warping (CDTW) is a more robust distance metric that improves upon DTW and Fréchet distances. Existing algorithms for CDTW are either exact algorithms that focus on non-Euclidean norms and piecewise linear curves, or approximation algorithms limited to piecewise linear curves. We present an approximation algorithm for computing the CDTW distance under Euclidean norm between piecewise (higher degree) algebraic curves. That is, we present a fully polynomial-time approximation scheme (FPTAS) of multiplicative error $\varepsilon$, with $O \left( (m+n)^{\frac{19}{6}} (\frac{1}{\varepsilon})^{\frac{10}{3}} \log \left( \frac{ (m+n) }{\varepsilon^2} \right) \right)$ complexity, where $m$ and $n$ are the number of pieces of the two input curves.

cs.CG↗

arXiv2026

The microscope is the mask: privileged views and labels from a cryo-ET forward model

We explore the use of simulated data for training a model for protein annotation in crowded cryo-electron tomography volumes reconstructed from images collected at limited tilt angles and severely corrupted by the measurement operator. Firstly, we leverage the corruptions imposed by the forward model to generate domain-specific augmented paired views of the exact same scene for an invariance objective integrated into the LeJEPA self-supervised training framework. Secondly, we use additional information from the simulation pipeline such as the positions and identity of proteins in the simulated volumes to inform the architecture of the model and the loss function, so that semantic information is localised at protein positions in the resulting dense feature volume. The resulting model, CARNIVAL, is evaluated without finetuning on classification and detection tasks in real tomograms, using a benchmark dataset containing multiple protein types and two tomogram processing types. We show that CARNIVAL outperforms a state-of-the-art model trained using a contrastive objective on simulated data but without forward model-based paired views or privileged information.

cs.CV↗

arXiv2026

Gradient Descent on Logistic Regression with Non-Separable Data and Large Step Sizes

We study gradient descent (GD) dynamics on logistic regression problems with large, constant step sizes. For linearly-separable data, it is known that GD converges to the minimizer with arbitrarily large step sizes, a property which no longer holds when the problem is not separable. In fact, the behaviour can be much more complex -- a sequence of period-doubling bifurcations begins at the critical step size $2/λ$, where $λ$ is the largest eigenvalue of the Hessian at the solution. Using a smaller-than-critical step size guarantees convergence if initialized nearby the solution: but does this suffice globally? In one dimension, we show that a step size less than $1/λ$ suffices for global convergence. However, for all step sizes between $1/λ$ and the critical step size $2/λ$, one can construct a dataset such that GD converges to a stable cycle. In higher dimensions, this is actually possible even for step sizes less than $1/λ$. Our results show that although local convergence is guaranteed for all step sizes less than the critical step size, global convergence is not, and GD may instead converge to a cycle depending on the initialization.

cs.LG↗

arXiv2026

Optimal Sensor and Actuator Selection for Factored Markov Decision Processes: Complexity, Approximability and Algorithms

Factored Markov Decision Processes (fMDPs) are a class of Markov Decision Processes (MDPs) in which the states (and actions) can be factored into a set of state (and action) variables and can be encoded compactly using a factored representation. In this paper, we consider a setting where the state of the fMDP is not directly observable, and the agent relies on a set of potential sensors to gather information. We formulate the problem of selecting a set of sensors for fMDPs (under a limited budget) to maximize the infinite-horizon discounted return provided by the optimal policy. We show the fundamental result that it is NP-hard to approximate this problem to within a factor of $n^{1-c}$ for any $c > 1$, where $n$ is the number of state variables. Our inapproximability results for sensor selection also extend to a general class of Partially Observable MDPs (POMDPs). We also consider the dual problem of budgeted actuator selection (at design-time) to maximize the expected return under the optimal policy, for which we establish similar inapproximability results. Finally, we consider a simple greedy algorithm and empirically show that, despite the lack of formal theoretical guarantees, it performs effectively in practice, achieving on average over $70\%$ of the optimal solution value across a variety of real-world and randomly generated problem instances.

eess.SY↗

arXiv2026

Sample Complexity of Linear Quadratic Regulator Without Initial Stability

Inspired by REINFORCE, we introduce a novel receding-horizon algorithm for the Linear Quadratic Regulator (LQR) problem with unknown dynamics. Unlike prior methods, our algorithm avoids reliance on two-point gradient estimates while maintaining the same order of sample complexity. Furthermore, it eliminates the restrictive requirement of starting with a stable initial policy, broadening its applicability. Beyond these improvements, we introduce a refined analysis of error propagation through the contraction of the Riccati operator under the Riemannian distance. This refinement leads to a better sample complexity and ensures improved convergence guarantees.

math.OC↗

arXiv2026

Quantum Speedups for Sampling and Non-convex Optimization with Stochastic Oracles

We present quantum speedups for sampling from distributions of the form $π\propto e^{-f}$ on $\mathbb{R}^d$. We consider two stochastic oracle models: a stochastic gradient oracle, where $f=\frac{1}{n}\sum_{i=1}^n f_i $ and component gradients $\{\nabla f_i\}_{i \in [n]}$ are available, and a stochastic evaluation oracle, where only noisy values of $f$ are available. Our framework accelerates classical stochastic Langevin Monte Carlo (LMC) and Hamiltonian Monte Carlo (HMC) algorithms by replacing stochastic gradient estimators with variance-controlled quantum mean estimation and gradient estimation subroutines. Unlike quantum walk based approaches, our algorithms do not require reversibility or exact gradients, and they preserve the structure of the underlying Markov chain. In the finite-sum setting, quantum mean estimation combined with classical variance-reduction techniques improves the stochastic gradient-query complexity for the approximate sampling task. In the stochastic zeroth-order setting, we develop gradient estimators robust to noisy function evaluations, yielding improved evaluation complexity for LMC and HMC. These results apply to strongly log-concave and/or non-log-concave distributions satisfying a log-Sobolev inequality, with convergence guarantees in Wasserstein distance and Kullback--Leibler divergence. We also show that faster sampling methods lead to quantum speedups for optimization, including for non-smooth and approximately convex objectives.

quant-ph↗

arXiv2026

Computable Characterisations of Scaled Relative Graphs of Closed Operators

The Scaled Relative Graph (SRG) is a promising tool for stability and robustness analysis of multi-input multi-output systems. In this paper, we provide tools for exact and computable constructions of the SRG for closed linear operators, based on maximum and minimum gain computations. The results are suitable for bounded and unbounded operators, and we specify how they can be used to draw SRGs for the typical operators that are used to model linear-time-invariant dynamical systems. For the special case of state-space models, we show how the Bounded Real Lemma can be used to construct the SRG. Furthermore, we give an example of how stability analysis of dynamical systems can be done with SRGs.

eess.SY↗

arXiv2026

An Energy-Based Mechanism for Compositional Behavior

Flexible intelligence relies on the ability to reuse previously acquired behaviors and combine them differently as circumstances change. In biological and artificial systems, this ability is often attributed to gating mechanisms that determine how much each available behavior should contribute at a given time. Yet these gating rules, the dynamics that compute them, and the neural circuits that may implement them are usually introduced separately, leaving unclear whether they reflect a common underlying principle. Here, we show that they can all be derived from a single variational principle for behavioral composition. The resulting mechanism naturally gives rise to softmax gating, evolves as an energy-based dynamical system with guaranteed convergence, and admits a recurrent neural network instantiation featuring context-dependent and local interactions. Across collective behavior, human decision-making, and layered control, the same mechanism reproduces characteristic behavioral patterns, provides interpretable accounts of how different behaviors are combined, and matches or outperforms established approaches. These results provide a unified account of how behavioral composition can emerge from a common principle, with implications for understanding flexible behavior in natural systems and for designing artificial agents that can adapt by recombining existing capabilities.

math.OC↗

arXiv2026

Constrained dynamics for searching saddle points on embedded Riemannian submanifolds of Euclidean space

Finding constrained saddle points on embedded Riemannian submanifolds of Euclidean space is significant for analyzing energy landscapes arising in physics and chemistry. Existing works exploit explicit global/local regular level-set representations of manifolds, which may be unavailable or computationally inconvenient for manifolds represented through, e.g., projectors, factorizations, or rank constraints. In this paper, we develop a constrained saddle dynamic based on embedded-submanifold geometric primitives, completely avoiding the use of explicit representations. In particular, our dynamic is formulated compactly on the Grassmann bundle of the tangent bundle. By analyzing the Grassmann bundle geometry, we rigorously establish the local linear stability of the dynamic and the local linear convergence of the resulting algorithms. Remarkably, our analysis provides the first iterate convergence result for discretized algorithms to saddle points of prescribed indices in embedded-submanifold settings. Moreover, by virtue of the Grassmann bundle formulation, we remove unnecessary nondegeneracy assumptions on the eigenvalues of the Riemannian Hessian that are present in existing works. We also point out that locating saddle points can be more ill-conditioned than finding local minimizers, and requires using nonredundant parametrizations. Finally, numerical experiments on linear eigenvalue problems and electronic excited-state calculations showcase the effectiveness of the proposed algorithms and corroborate the established local theory.

math.NA↗

arXiv2026

Trajectory-Wise Certification for Vector-Field Mirror Descent with Deterministic Finite Differences

We study mirror descent in which the objective gradient is replaced by a general vector field. Since such a field does not automatically relate the mirror update to the objective gap, we introduce a trajectory-wise generalized relative-smoothness condition and a generalized star-convexity interface. Together they yield a finite-horizon, a posteriori last-iterate certificate with an accumulated-stepsize term and an exceptional-region-dependent error term. We also give a pointwise sufficient condition for positive admissible stepsizes, displaying the effects of objective smoothness, mirror-map conditioning, and vector-field mismatch. We then construct a deterministic zeroth-order instance from coordinate central differences. The function values place the unknown gradient in an explicit uncertainty box, and verification of the interface reduces to robust conic dominance. We derive an explicit scaling of the central-difference vector that guarantees the required dominance over the gradient uncertainty set. Under uniform Hessian bounds, the resulting vector field satisfies the interface outside an explicit resolution-dependent neighborhood. The final guarantee combines a rate term governed by accumulated accepted stepsizes with a finite-resolution error floor.

math.OC↗

arXiv2026

Variational Contraction Conditions for Iterative Algorithms in Multi-Population Discrete-Time Regularized Mean-Field Games

In this work, we study the contraction conditions of iterative algorithms for stationary and finite-horizon discrete-time regularized mean-field games (MFGs) with multiple populations, where each population only interacts with the state distributions of the other populations. Due to the high dimensionality caused by the interaction of different populations, contraction rates for these algorithms cannot, in general, be expressed in terms of radicals. By studying the dynamics of these iterative algorithms and assuming that the system components of each population's MFG are Lipschitz continuous, we present explicit (eventual) contraction conditions for each algorithm in any normed space, relying only on these Lipschitz parameters. As a consequence of these contraction conditions, we provide convergence rates of finite-horizon mean-field equilibria to infinite-horizon stationary (and non-stationary) mean-field equilibria (MFEs), under restrictions on a variational characterization of the dynamics of these iterative algorithms. In the single-population case, the restrictions we impose on this variational characterization to obtain these convergence results are less restrictive than previous results in the literature.

math.OC↗

arXiv2026

Scalable Self-Supervised Learning for Multiphase AC-OPF in Distribution Systems with Topology Reconfiguration

The proliferation of distributed energy resources (DERs) in distribution grids enables the active coordination of these assets to reduce costs and enable cleaner operations. Realizing this potential requires solving multiphase AC optimal power flow (AC-OPF) quickly across varying loads, DER availabilities, and topology reconfigurations, at much greater speed and scale than conventional nonlinear solvers. Learning-based surrogates can offer millisecond inference, yet existing methods target largely balanced transmission systems and do not scale to the multiphase, unbalanced, and reconfigurable nature of distribution feeders at utility scale. We present the Penalty + Sequential Linearized Feasibility Seeking (SLFS) algorithm, a self-supervised learning framework for multiphase distribution AC-OPF under switch-induced topology changes. Penalty+SLFS requires no labeled optimal solutions and trains directly from the AC-OPF objective and constraints through a differentiable fixed-point power flow solver, avoiding expensive label generation and admitting robust training procedures. Topology changes are handled efficiently using Sherman-Morrison-Woodbury updates of the admittance-matrix inverse, while an M-step Jacobian approximation accelerates differentiation through the power flow solver. At inference, SLFS repairs any infeasible predictions, providing feasibility guarantees with low computational overhead. On IEEE feeders ranging from 13 to 8,500 nodes, Penalty+SLFS achieves negligible optimality gaps and near-zero constraint violations, delivers up to three orders of magnitude speedups over IPOPT, and remains robust under large distributional shifts, demonstrating a viable path toward real-time, topology-aware AC-OPF for large-scale distribution grids.

eess.SY↗

arXiv2026

Turnpike properties in nonlinear system identification

We analyze the problem of learning general discrete-time nonlinear state-space models using the simulation error minimization (SEM) method. In this setting, model parameters are typically learned by minimizing the mismatch between simulated and measured outputs over a training dataset, or shorter subsequences extracted from it. Specifically, we study the cumulative output turnpike property of the underlying SEM optimization problem, which requires optimal output sequences emanating from a fixed initial state to approach and remain close to an optimal output sequence of the corresponding SEM problem with free initial state. In the presence of non-unique optimal output sequences---as may arise, for instance, in fully black-box system identification using neural networks---the property is formulated with respect to the closest such sequence. Turnpike behavior is generally desirable in practice, as it provides a theoretical justification for employing computationally more tractable SEM formulations with fixed initial states while ensuring that their optimal output sequences remain close to unconstrained optimal ones. Under a mild reachability condition, we establish equivalence between the cumulative turnpike property, coercivity of the value function, and a tailored notion of strict dissipativity. We additionally introduce a cardinality turnpike property and show that it is strictly weaker than the cumulative notion. Finally, we establish sufficient conditions for turnpike behavior based on incremental output stability, convexity of the stage cost, and a suitable optimality condition, and illustrate the theory by means of a numerical example.

eess.SY↗

arXiv2026

Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence

The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.

math.NA↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Projected Riemannian Gradient Descent for the Bures-Wasserstein Barycenter: Dimension-Independent Linear Convergence at Unit Step Size	2026-09-03	2609.03762	arxiv
Adapting to Evolving Requirements: Agentic AI for Retail Supply Chain Operations	2026-09-03	2609.03860	arxiv
Transversality Conditions for Boundary Constraints Defined by Differential Equations	2026-09-03	2609.04084	arxiv
Minimizing the makespan in job shop scheduling under conflict graph constraints	2026-10-15	2609.04161	arxiv
Approximating CDTW Distance of Piecewise Algebraic Curves	2026-09-03	2609.04294	arxiv
The microscope is the mask: privileged views and labels from a cryo-ET forward model	2026-09-03	2609.04325	arxiv
Gradient Descent on Logistic Regression with Non-Separable Data and Large Step Sizes	2026-09-02	2406.05033	arxiv
Optimal Sensor and Actuator Selection for Factored Markov Decision Processes: Complexity, Approximability and Algorithms	2026-09-02	2407.07310	arxiv
Sample Complexity of Linear Quadratic Regulator Without Initial Stability	2026-09-02	2502.14210	arxiv
Quantum Speedups for Sampling and Non-convex Optimization with Stochastic Oracles	2026-09-02	2504.03626	arxiv
Computable Characterisations of Scaled Relative Graphs of Closed Operators	2026-09-02	2511.08420	arxiv
An Energy-Based Mechanism for Compositional Behavior	2026-09-02	2512.04745	arxiv
Constrained dynamics for searching saddle points on embedded Riemannian submanifolds of Euclidean space	2026-09-02	2601.03931	arxiv
Trajectory-Wise Certification for Vector-Field Mirror Descent with Deterministic Finite Differences	2026-09-02	2602.00634	arxiv
Variational Contraction Conditions for Iterative Algorithms in Multi-Population Discrete-Time Regularized Mean-Field Games	2026-09-02	2605.23906	arxiv
Scalable Self-Supervised Learning for Multiphase AC-OPF in Distribution Systems with Topology Reconfiguration	2026-09-02	2608.25095	arxiv
Turnpike properties in nonlinear system identification	2026-09-02	2609.02071	arxiv
Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence	2026-09-02	2609.02072	arxiv

These are bibliographic comparisons, not experimental rankings. Follow the original document for methods and conditions.