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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

Race, Exchange, Improve: Finding high-quality MIP solutions quickly

Mixed-integer programming (MIP) is a cornerstone in applied optimization, both in industry and academia. Recently, there has been increased attention to finding strong primal solutions quickly. This is reflected, for example, in the development of the NVIDIA cuOpt solver and, most recently, in the new MIPFEAS benchmark, which has a tight time limit of 600 seconds and evaluates solvers based on how quickly they find high-quality primal solutions. This article introduces a MIP portfolio parallelization scheme, focusing on efficiently exchanging information between its workers. We present two implementations of this scheme: one built directly into the open-source MIP solver SCIP, and an external one, which we call ReXi. ReXi is currently the fastest non-commercial solver in the MIPFEAS benchmark, followed by the SCIP-integrated implementation. Moreover, we present new versions of both implementations that considerably outperform their predecessors on the MIPFEAS benchmark.

math.OC↗

arXiv2026

The Role of Gradient Modification in Heavy-Tailed Nonconvex Stochastic Min-Max Optimization

Stochastic min-max optimization has attracted increasing attention due to its applications in modern machine learning, while existing theoretical studies mainly rely on the bounded variance assumption for stochastic gradients. Under heavy-tailed noise, where stochastic gradients only possess a finite $p$-th moment for $p\in(1,2]$, gradient clipping or normalization is commonly believed to be necessary to guarantee convergence. In this work, we revisit stochastic min-max optimization under heavy-tailed noise and provide a comprehensive theoretical study of stochastic gradient descent ascent (SGDA). We first show that vanilla SGDA, without any modification to its update rule, can converge under heavy-tailed noise in both nonconvex-strongly-concave (NC-SC) and nonconvex-concave (NC-C) settings, establishing the first convergence guarantees for SGDA in these regimes. Beyond unregularized problems, we further investigate regularized stochastic min-max optimization, where directly incorporating gradient normalization into proximal updates is nontrivial due to the incompatibility between normalization and proximal structures. We overcome this difficulty by developing new clipping-free algorithms, i.e., Stoc-TRGDAM and Stoc-TRGDmax, and they both can achieve the optimal dependence on the target accuracy without using gradient clipping.

math.OC↗

arXiv2026

Low-rank matrix recovery landscapes beyond RIP with application to rank-one measurements

We study the problem of low-rank matrix recovery from linear measurements via the global nonconvex landscape of a low-rank factored formulation of the matrix LASSO (nuclear-norm--regularized least-squares). If the landscape is benign, that is, has no bad local optima, then practical and scalable algorithms can compute good statistical estimates. Previous state-of-the-art landscape guarantees have typically assumed that the linear measurement operator has the restricted isometry property, that is, the operator is approximately an isometry over all low-rank matrices. This is an unrealistic assumption for many applications; in particular, when the individual measurement matrices are themselves low-rank, we typically have poor upper isometry constants. To overcome this, we establish new guarantees of a benign landscape under a weaker isometry condition: rather than requiring upper isometry over all low-rank matrices, we only require it over the linear low-rank tangent space to the low-rank ground truth matrix. To illustrate the utility of this result, we apply it to the problem of matrix recovery from random rank-one linear measurements; via high-probability concentration bounds on the random measurement operator, we prove a novel landscape guarantee with statistically near-optimal sample complexity and recovery error.

math.OC↗

arXiv2026

Recovering linear images of sparse signals from indirect observations

In this paper, we develop and analyze techniques for recovering a linear image $Bx$ of an unknown signal $x$ from indirect noisy observation $ω=Ax+ξ$. It is {\em a priori} known that $x\in \cX$, a given convex compact set, and that $x$ is $s$-sparse---has at most $s$ nonvanishing entries. The proposed estimates belong to a large family of recovery routines by $\ell_1$-minimization. However, unlike the classical result describing performance of such estimates, we do not make any special (and hard to check) assumptions about the sensing matrix $A$ such as nullspace or Restricted Isometry condition and the like. As a consequence, parameters of the estimates and the upper bounds on their risks are not available in a closed analytic form, but are delivered instead by efficient computation as solutions to explicit convex optimization problems.

stat.ML↗

arXiv2026

Fast PAC Global Optimization via Restarted Langevin: Exploration, Exploitation, and Degenerate Cooling

This paper addresses a fundamental question in non-convex optimization: \textit{How should a stochastic optimizer allocate computation between global exploration and local exploitation?} We study this question in continuous time, using Langevin dynamics for global exploration and deterministic gradient flow for local exploitation. The simplest algorithm in this class is the best-state Langevin--gradient method: several Langevin trajectories explore the objective landscape, the best state encountered is retained, and a single gradient trajectory then refines this state to high terminal accuracy. Our main objective is to characterize the computational work required to achieve a prescribed accuracy with prescribed confidence. We develop low-temperature approximations for this probably approximately correct (PAC) work--accuracy tradeoff, both for the best-state method and for related Langevin schemes. Global exploration is governed by energy barriers, through spectral-gap and Eyring--Kramers asymptotics. The resulting approximations expose the competing effects of temperature, computational budget, and the number of exploratory trajectories, and quantify their dependence on the confidence parameter. In contrast, terminal accuracy is largely separated from global exploration: once a suitable region of attraction has been reached, gradient flow requires only $O(\log(1/\varepsilon))$ additional work to reach accuracy at least $\varepsilon$. Numerical experiments on challenging objectives serve to compare algorithms and to test the quantitative predictions of the asymptotic theory. They identify regimes in which the low-temperature approximations provide useful guidance, as well as regimes exhibiting substantial temperature sensitivity and potentially significant high-dimensional limitations.

math.OC↗

arXiv2026

Weak Convexity and Proximal Bundle Methods for Nonsmooth Policy Optimization in Robust Control

We study policy optimization for discrete-time robust $\mathcal{H}_\infty$ control with static output-feedback, and present the first feasibility-preserving algorithm with a deterministic, non-asymptotic complexity guarantee. This problem naturally leads to a nonsmooth and nonconvex optimization over the set of stabilizing feedback gains. We first establish several structural properties of the $\mathcal{H}_\infty$ cost. In particular, we show that the cost is weakly convex on every convex subset of a sublevel set. For the state-feedback case, we further establish a weak Polyak--Łojasiewicz inequality, which ensures that every stationary point is globally optimal. Building on these properties, we develop a proximal bundle method for $\mathcal{H}_\infty$ policy optimization. The proposed method can be viewed as an implementable approximation of the proximal point method and uses only function value and subgradient information. We show that all iterates remain stabilizing and establish a deterministic non-asymptotic complexity bound of $\mathcal{O}(\max\{η^{-4},ε^{-2}\})$ for finding an $(η,ε)$-stationary point. Numerical experiments illustrate our theoretical results.

math.OC↗

arXiv2026

Riccati Stability Without Auxiliary Matrices

In the 2004 collection \emph{Unsolved Problems in Mathematical Systems and Control Theory}, Erik Verriest posed the problem of characterizing Riccati stability ``without invoking additional matrices.'' We give such a characterization through a scalar invariant of the resolvent family. The invariant is defined by covariance balances using at most $n^2+1$ frequency--direction pairs. A finite-dimensional separation argument shows that this covariance radius equals the optimal common ellipsoidal norm of the resolvent family. Combined with the strict bounded real lemma, this yields a necessary and sufficient condition for Riccati stability involving only the Hurwitz property of the first matrix and a single scalar inequality. The invariant reduces to the ordinary spectral radius for one matrix, lies between the pointwise spectral-radius and unscaled small-gain levels of the resolvent family, and coincides with the operator-space spectral radius of Shalit and Shamovich for the natural resolvent function space. The main results have been formally verified in the Lean proof assistant.

math.OC↗

arXiv2026

Robust conditional dimension reduction for dissimilarity data

Conditional dimension reduction (cDR) learns low-dimensional latent coordinates while accounting for observed covariates that represent known sources of variation in the data. Conditional Multidimensional Scaling (cMDS) is a cDR technique that works directly with dissimilarity data. Its standard squared-stress formulation, however, is sensitive to contaminated dissimilarity, since outliers can dominate the objective and distort the learned configuration. We proposed Robust Conditional Multidimensional Scaling (rcMDS) by replacing the squared-stress criterion with a Fair M-estimation objective. We developed a reweighted conditional SMACOF algorithm to optimize this objective. The proposed algorithm admits computationally tractable updates, and its stabilized objective values decrease monotonically and converge to a finite limit. Experiments on synthetic and real data show that the pro

stat.ML↗

arXiv2026

A robust and adaptive MPC formulation for Gaussian process models

In this paper, we present a robust and adaptive model predictive control (MPC) framework for uncertain nonlinear systems affected by bounded disturbances and unmodeled nonlinearities. We use Gaussian Processes (GPs) to learn the uncertain dynamics based on noisy measurements, including those collected during system operation. As a key contribution, we derive robust predictions for GP models using contraction metrics, which are incorporated in the MPC formulation. The proposed design guarantees recursive feasibility, robust constraint satisfaction and convergence to a reference state, with high probability. We provide a numerical example of a planar quadrotor subject to difficult-to-model ground effects, which highlights significant improvements achieved through the proposed robust prediction method and through online learning.

eess.SY↗

arXiv2026

Exact affine conditioning beyond Gaussians: a unique characterization of the ensemble Kalman update

The analysis step of the stochastic ensemble Kalman filter, called the ensemble Kalman update (EnKU), is widely used for approximating posterior distributions in inverse problems and data assimilation. The EnKU approximates the posterior distribution $π_{X\mid Y=y_\star}$ by pushing forward the joint distribution $(X,Y)\simπ$ through an affine map $L^{\mathrm{EnKU}}_{π,y_\star}(x,y)$ that depends only on the covariance structure of $π$ and the observation $y_\star$. While the EnKU yields the exact posterior for Gaussian $π$ in the mean-field, this property alone does not uniquely determine the EnKU. In fact, there are infinitely many affine maps $L_{π, y_\star}$ that achieve such exact conditioning. In this paper, we offer a novel characterization of the EnKU among all such affine maps. We first exhaustively characterize the set ${E}^{\mathrm{EnKU}}$ of joint distributions for which the EnKU yields exact conditioning, showing that it is much larger than the set of Gaussians. Next, we show that except for a small class of highly symmetric distributions within ${E}^{\mathrm{EnKU}}$, the EnKU is the {unique} exact affine conditioning map. Further, we characterize the largest possible set of distributions ${F}$ for which a distribution-dependent, weakly observation-dependent, affine map exists, a class of transports that naturally includes the EnKU. We show that ${F}={E}^{\mathrm{EnKU}}\cup{S}_{\mathrm{nl-dec}}$ with a small symmetry class ${S}_{\mathrm{nl-dec}}$, meaning that for affine conditioning beyond the Gaussian setting, the EnKU has an exact set that is essentially maximally large.

math.ST↗

arXiv2026

Generalized infinite dimensional Alpha-Procrustes based geometries

This work extends the recently introduced Alpha-Procrustes family of Riemannian metrics for symmetric positive definite (SPD) matrices by incorporating generalized versions of the Bures-Wasserstein (GBW), Log-Euclidean, and Wasserstein distances. While the Alpha-Procrustes framework has unified many classical metrics in both finite- and infinite- dimensional settings, it previously lacked the structural components necessary to realize these generalized forms. We introduce a formalism based on unitized Hilbert-Schmidt operators and an extended Mahalanobis norm that allows the construction of robust, infinite-dimensional generalizations of GBW and Log-Hilbert-Schmidt distances. Our approach also incorporates a learnable regularization parameter that enhances geometric stability in high-dimensional comparisons. Preliminary experiments reproducing benchmarks from the literature demonstrate the improved performance of our generalized metrics, particularly in scenarios involving comparisons between datasets of varying dimension and scale. This work lays a theoretical and computational foundation for advancing robust geometric methods in machine learning, statistical inference, and functional data analysis.

stat.ML↗

arXiv2026

Centered Permutation Prefixes for SGD with Random Reshuffling: Sharp Rates, Hölder Geometry, and Composite Proximal Extensions

We study stochastic gradient descent with random reshuffling for finite sums \[ F(x)=\frac1n\sum_{i=1}^n f_i(x). \] For fresh reshuffling with a constant component stepsize, if each $f_i$ has an $L$-Lipschitz gradient and the average $F$ is $μ$-strongly convex with a Lipschitz-continuous Hessian, we prove the last-epoch rate \[ \mathbb E[F(y_K)-F(x_\star)] =\widetilde O\!\left(T^{-2}+n^2T^{-3}\right), \qquad T=nK, \] matching the known quadratic lower bound in its $(n,K)$-dependence. The components may be nonconvex, and no componentwise Hessian continuity or separate bounded-iterate assumption is required. More generally, a $ν$-Hölder-continuous average Hessian adds only $\widetilde O(n^{1+ν}T^{-2-2ν})$, so every $ν\ge 1/2$ preserves the quadratic rate. Under convex components, a decreasing-stepsize result removes the large-epoch requirement and recovers the same two-term scale once $nK$ exceeds the condition-number scale. We also analyze epoch-wise ProxRR for $\mathcal P=F+ψ$. Writing $x^\dagger$ for the composite minimizer and $β_\star=\|\nabla F(x^\dagger)\|$, we prove \[ \mathbb E\|y_K-x^\dagger\|^2 =\widetilde O\!\left( \frac{β_\star^2}{K^2} +T^{-2}+n^2T^{-3} +n^{1+ν}T^{-2-2ν} \right). \] For $ν\ge 1/2$, we show that the $β_\star^2/K^2$ splitting term is unavoidable and obtain a matching lower bound up to logarithms in the stated constant-stepsize regime.

math.OC↗

arXiv2026

Blind Random Search with Noisy Loss Measurements: Averaging, Thresholding, and Almost Sure Convergence

Blind random search repeatedly draws a candidate point and replaces the current estimate whenever the candidate has a lower loss. In the absence of noise, the true loss is observed directly. It decreases strictly at every accepted update and is monotone nonincreasing over all iterations. Measurement noise can make a worse candidate appear better and thereby break this monotonicity. To recover almost sure convergence under noise, we incorporate averaging and thresholding into the original decision criterion. These two classical tools are coupled. As the sample sizes grow, the positive threshold shrinks at a matched rate. These modifications allow blind random search to recover eventual monotonicity of the true loss under noisy measurements and to converge almost surely.

math.OC↗

arXiv2026

On the differential of the exponential map

We study the time derivative of the matrix exponential $B(t)=\mathrm{exp}(A(t))$, where $A(t)$ is a time-parametrized curve in $\mathrm{Mat}(n)$. Starting from the Taylor series expansion, we derive a nested summation formula, which is then reformulated into a double summation. This expression is converted into an integral representation using Euler's beta function, and further expressed in terms of Lie brackets and the adjoint representation. Later, we extend this to a family of problems and verify a well-known result. Finally, we establish connections to the Gateaux and Fréchet derivatives, proving the latter's existence. These results offer a unified and explicit framework for understanding the derivative of the exponential map.

eess.SY↗

arXiv2026

Resilience Beyond Stationary Client Unavailability: Unlocking Efficient and Unbiased Federated Learning

Due to resource constraints or external and internal uncertainties, clients in real-world federated learning systems are often intermittently available edge devices. In highly dynamic environments, the parameter server lacks prior real-time knowledge of clients' availability, making it challenging to adapt traditional federated learning algorithms to be resilient to uncertainties in client availability. If not carefully addressed, complex client availability can introduce significant bias, potentially harming the performance of the trained model. Most prior work either fails to account for non-stationary client availability dynamics or demands significant memory and computational overhead. This paper aims to develop efficient federated learning algorithms that are provably resilient to heterogeneous and non-stationary stochastic client availability. We propose FedSWE, which admits novel algorithmic structures to (i) compensate for missed computations, (ii) stabilize and diffuse the global updates over rounds, and (iii) evenly mix the local updates through implicit gossiping, despite being agnostic to non-stationary dynamics. Compared with the standard FedAvg, FedSWE introduces light additional memory and computation overhead. We show that FedSWE converges to a stationary point of non-convex objectives while achieving the desired linear speedup property in certain special cases. We corroborate our analysis with numerical experiments over diversified client unavailability dynamics on real-world data sets.

cs.LG↗

arXiv2026

Near-Field Physical-Layer Authentication Under Impersonation Attacks

This paper studies physical-layer authentication (PLA) in the near-field regime under impersonation attacks. Unlike the far-field case, where the steering vector depends only on the angle of arrival (AoA), in the near field it depends on both angle and distance. We analyze the attack by minimizing the mean-square error (MSE) between the signal received from a legitimate transmitter Alice and the signal generated by an active attacker Eve. For a single-antenna Eve, we derive the optimal scalar precoder and show that, for an inter-element spacing no larger than half a wavelength and under the standard second-order Fresnel approximation, perfect impersonation is possible only if Eve has the same angle and the same distance from Bob as Alice. We then extend the analysis to a multi-antenna Eve and derive the optimal precoding vector. In this case, perfect impersonation is possible only if Alice steering vector belongs to the subspace spanned by Eve steering vectors. Simulation results show that, in the near field, a distance difference is sufficient to prevent a successful impersonation attack even when Alice and Eve have the same AoA, and confirm the analytical results.

cs.IT↗

arXiv2026

On the Delay-Constrained Maximum Concurrent Flow Problem

Real-time services, such as VoIP and large-scale neural network training, require strict transmission delay guarantees. While routing under hop constraints is tractable, real-world delays increase sharply with equipment load, typically modeled using the M/M/1 queuing function where delay is inversely proportional to available bandwidth. We investigate the resulting Delay-Constrained Maximum Concurrent Flow (DCMCF) problem, which seeks to maximize the minimum throughput across all commodities. The problem's complexity stems from the conditional and non-linear nature of the delay constraints, which are active only along the specific paths used by the flow. We prove that DCMCF is strongly NP-hard, even for single-source/single-destination instances. To address the inherent non-convexity of the problem, we introduce a new convex relaxation expressed through second-order cone constraints, obtained from the convex envelope of a function representing the conditional delay associated with a single arc of a given path. The relaxation is shown to outperform existing formulations based on disjunctive programming. Leveraging this result, we develop a polynomial-time approximation algorithm with a provable performance guarantee and present numerical experiments demonstrating the effectiveness of the proposed approach.

cs.NI↗

arXiv2026

Quantum lower bounds for convex optimization and real matrix-vector query problems

We (the author and the AI systems that did the heavy lifting) show that the quantum query complexity of minimizing a convex function over a convex subset of $\mathbb{R}^n$ with evaluation and membership queries is $\tildeΩ(n)$, nearly matching the best known upper bound. In particular, we show this even for quadratic minimization, which is equivalent to inverting an $n \times n$ real matrix using matrix-vector queries. We also show linear or nearly linear lower bounds on the quantum query complexity of computing the trace, the sign of the determinant, and the magnitude of the determinant of a real matrix in the matrix-vector query model. We use a novel quantum lower bound technique, the determinantal witness method, based on identifying a witness whose Fourier transform vanishes on low-rank matrices and that correlates well with the function being computed.

quant-ph↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Race, Exchange, Improve: Finding high-quality MIP solutions quickly	2026-09-05	2609.05954	arxiv
The Role of Gradient Modification in Heavy-Tailed Nonconvex Stochastic Min-Max Optimization	2026-09-05	2609.06064	arxiv
Low-rank matrix recovery landscapes beyond RIP with application to rank-one measurements	2026-09-05	2609.06146	arxiv
Recovering linear images of sparse signals from indirect observations	2026-09-05	2609.06182	arxiv
Fast PAC Global Optimization via Restarted Langevin: Exploration, Exploitation, and Degenerate Cooling	2026-09-28	2609.06196	arxiv
Weak Convexity and Proximal Bundle Methods for Nonsmooth Policy Optimization in Robust Control	2026-09-05	2609.06202	arxiv
Riccati Stability Without Auxiliary Matrices	2026-10-10	2609.06275	arxiv
Robust conditional dimension reduction for dissimilarity data	2026-09-05	2609.06284	arxiv
A robust and adaptive MPC formulation for Gaussian process models	2026-09-04	2507.02098	arxiv
Exact affine conditioning beyond Gaussians: a unique characterization of the ensemble Kalman update	2026-09-04	2510.00158	arxiv
Generalized infinite dimensional Alpha-Procrustes based geometries	2026-09-04	2511.09801	arxiv
Centered Permutation Prefixes for SGD with Random Reshuffling: Sharp Rates, Hölder Geometry, and Composite Proximal Extensions	2026-09-04	2609.04578	arxiv
Blind Random Search with Noisy Loss Measurements: Averaging, Thresholding, and Almost Sure Convergence	2026-09-04	2609.04636	arxiv
On the differential of the exponential map	2026-09-04	2609.04762	arxiv
Resilience Beyond Stationary Client Unavailability: Unlocking Efficient and Unbiased Federated Learning	2026-09-04	2609.04763	arxiv
Near-Field Physical-Layer Authentication Under Impersonation Attacks	2026-09-18	2609.04879	arxiv
On the Delay-Constrained Maximum Concurrent Flow Problem	2026-09-04	2609.05068	arxiv
Quantum lower bounds for convex optimization and real matrix-vector query problems	2026-09-04	2609.05679	arxiv

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