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arXiv2026

A 5/3 Guarantee for Echelon Stock (R,nQ) Policies in Two-Stage Stochastic Serial Systems

Fixed shipment costs encourage large orders, while inventory and customer backlog costs favor frequent replenishment. In a serial system, the stages must also coordinate their lot sizes and material availability. We study the classical echelon stock $(R,nQ)$ policy: the downstream stage requests a fixed lot, the upstream stage orders an integer multiple of that lot, and transfers contain complete lots. For unit Poisson demand, deterministic lead times, setup and holding costs at both stages, and full backlogging, we prove that the optimized cost in this class is at most $5/3\approx1.667$ of the optimal admissible system cost, uniformly over all nonnegative cost rates, demand rates, and lead times. The guarantee holds even when the selected policy pays a setup charge for every lot while the optimal benchmark pays only once per aggregate shipment. To obtain the guarantee, we use a common inventory calculation that allows independently chosen replenishment frequencies while respecting upstream material availability. We compare its optimized value, an upper bound on the best integer-ratio policy cost, with a novel lower bound on the optimal system cost that combines allocated single-stage costs with the minimum inventory cost when setup costs are zero. We also show that no uniform factor below $(1+\sqrt{3})/2\approx1.366$ is possible for the classical class. Existing numerical experiments place best-found classical policies within $22.7\%$ of an evaluated lower bound across three broad grids. The results quantify both the reliability and the limits of a simple operating rule and explain how shipment consolidation, replenishment timing, and flexibility affect its value.

math.OC↗

arXiv2026

Parameter-Free Dynamic Regret under Heavy-Tailed Noise

We study online convex optimization with one unbiased stochastic subgradient per round and noise having a finite $p$-th central moment, where $p\in(1,2]$ is unknown. For a bounded convex domain of diameter $D$, subgradients bounded by $G$, noise scale $σ$, and comparator path length $P_T$, let $Λ_T=1+P_T/D$. A single algorithm, using none of $G,σ,p,P_T$, attains expected dynamic regret $O_p\left(\min\{GD\sqrt{TΛ_T}+σDT^{1/p}Λ_T^{(p-1)/p},\,GDT\}\right)$ against every fixed comparator sequence. Restarted AdaGrad experts produce the noise-path exponent $(p-1)/p$, and a prior favoring longer restart intervals removes horizon-dependent logarithmic overhead. We give an explicit bound uniform in $p$; its logarithm-free form has noise coefficient $O(1+\log(p/(p-1)))$, while the static-regret constant is universal. The analysis requires only marginal noise moments and permits dependent errors. Complete pathwise proofs retain both the expert-loss range and the gradient energies preceding comparator movement. Matching lower bounds hold on every bounded convex domain of positive diameter, under the same gradient-only information model. Together with a path-budget-tuned upper bound, they characterize the minimax rate with universal constants, including its linear-regret saturation.

cs.LG↗

arXiv2026

Accelerated primal-dual dynamics and algorithms for convex optimization with nonlinear inequality constraints

We consider convex optimization with nonlinear inequality constraints and develop a primal-dual multiplier framework that is consistent in continuous and discrete time. We first propose continuous-time dynamics with Nesterov-type vanishing damping $α/t$, together with compatible extrapolations of the dual variable and the nonlinear constraint mapping. Under convexity assumptions and $α\geq3$, we establish $\mathcal O(t^{-2})$ convergence rates for both nonlinear feasibility and the objective residual. In the noncritical regime $α>3$, with an admissible choice of the extrapolation parameter, we further prove that the entire primal-dual trajectory converges to a KKT pair and sharpen both continuous-time estimates to $o(t^{-2})$. We then derive an inexact accelerated primal-dual algorithm through a compatible discretization of a perturbed version of the dynamic. For composite convex objectives, a weighted summability condition on the primal inexactness yields $\mathcal O(k^{-2})$ rates for feasibility and the objective residual. In the corresponding noncritical regime, the discrete primal-dual sequence converges to a KKT pair and both residual estimates improve to $o(k^{-2})$. Thus the continuous and discrete results exhibit matching accelerated rates and matching asymptotic improvements. To the best of our knowledge, this is the first Nesterov-type primal-dual multiplier framework for convex optimization with nonlinear inequality constraints.

math.OC↗

arXiv2026

Exact-Form Regret for Gradient Descent, Mirror Descent and Follow-the-Regularized-Leader

Online gradient descent is usually studied through external regret, where the learner competes with fixed alternatives. Recent work shows that first-order methods control richer action-dependent deviations. We ask for a geometric characterization of the deviations with respect to which online gradient descent, mirror descent, and follow-the-regularized-leader (FTRL) achieve no regret. We identify exactness as the common principle. Exactness means that the relevant displacement field is generated by a scalar potential, or equivalently that the associated one-form is exact in the geometry used by the algorithm. This geometry depends on the algorithm. For gradient descent it is Euclidean geometry, for mirror descent it is the geometry induced by the regularizer, and for FTRL it is the cumulative dual state. Under mild regularity conditions, exactness yields sublinear regret, while nonzero circulation provides the complementary obstruction and leads to linear regret. This gives a unified geometric framework for understanding the deviation classes controlled by these algorithms and reveals that different first-order methods can control genuinely different classes of deviations. These deviation classes have direct consequences for learning, particularly in games. We study the equilibrium notions induced by exact-form deviations and introduce conservative correlated equilibrium, reflecting both the conservative geometry of the underlying displacement fields and the restricted family of deviations available to the players. We characterize its relation to correlated equilibrium, determine when the resulting equilibrium notions coincide and when they separate, and show how these relationships depend on the geometry and the learning algorithm. Overall, this work gives a unified geometric account of what first-order online learning algorithms are no-regret with respect to, beyond fixed comparators.

cs.LG↗

arXiv2026

Input-to-State Stability Framework for Fully Distributed Primal-Dual Dynamics for Quadratic GNEPs Without Multiplier Consensus

Generalized Nash Equilibrium Problems (GNEPs) often arise in multi-agent engineering applications that require distributed algorithms. Unlike traditional approaches that enforce consensus on multipliers, our method removes the need to share multipliers, reducing communication and improving privacy. As a result, different initializations can lead to different GNEs, including non-variational ones. We establish convergence under sufficient conditions using an input-to-state stability (ISS) framework.

math.OC↗

arXiv2026

Infrequent Resolving Algorithm for Online Linear Programming

Online linear programming (OLP) has gained significant attention from both researchers and practitioners due to its extensive applications such as online auctions, network revenue management, order fulfillment and advertising. Existing OLP algorithms fall into two categories: LP-based algorithms and LP-free algorithms. The former typically guarantees better performance but requires solving a large number of LPs, which could be computationally expensive. In contrast, LP-free algorithms only require first-order computations but induce a worse performance. In this work, we bridge the gap between these two extremes by proposing a well-performing algorithm that solves LPs at a few selected time points and conducts first-order computations at other time points. Specifically, for the case where the inputs are drawn from an unknown finite-support distribution, the proposed algorithm achieves a constant regret (even for the hard "degenerate" case) while solving LPs only $O(\log\log T)$ times over the time horizon $T$. Moreover, when we are allowed to solve LPs only $M$ times, we design the corresponding schedule such that the proposed algorithm can guarantee a nearly $O\left(T^{(1/2)^{M-1}}\right)$ regret. Our work highlights the value of resolving both at the beginning and the end of the selling horizon, and provides a novel framework to prove the performance guarantee of the proposed policy under different infrequent resolving schedules. Numerical experiments are conducted to demonstrate the efficiency of the proposed algorithms. The code is available at https://github.com/guokai-li/infrequent_resolving_olp.

cs.DS↗

arXiv2026

Communication-efficient ADMM over Hierarchical Networks

This paper develops a novel distributed optimization algorithm based on the Alternating Direction Method of Multipliers (ADMM) to solve hierarchical optimization problems over tree-structured networks, termed hierarchical ADMM (hADMM), with a particular focus on enhancing communication efficiency across the network. By rearranging the augmented Lagrangian to establish a query-response communication mechanism between nodes that explicitly exploits the hierarchical tree structure, the proposed algorithm significantly reduces communication costs compared to existing ADMM-based methods for hierarchical optimization. Furthermore, hADMM guarantees asymptotic convergence under convexity assumptions. We also present a convergence rate analysis based on linear matrix inequalities to characterize the maximum theoretically achievable convergence rates across different network topologies, showing that the proposed hADMM attains linear convergence under mild conditions. Three numerical experiments demonstrate that hADMM is compatible with arbitrary tree network structures and outperforms existing approaches in terms of communication efficiency.

math.OC↗

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

Sparse Data Augmentation for Optimization with Provable Guarantees

In nonconvex optimization problems arising in geometric machine learning, data augmentation is commonly used to promote invariance by averaging empirical losses over transformations of the data. Computing the fully augmented objective, however, requires access to every element of the transformation group $G$, which may be prohibitively expensive when $G$ is large or accessible only through sampling. We study whether full augmentation can instead be approximated using a small, fixed sample of transformations acquired before optimization and reused thereafter. Under suitable regularity conditions, we show that, with probability at least $1-δ$, gradient descent (GD) on the resulting sparsely augmented objective returns an $\varepsilon$-stationary point of the fully augmented objective using $\mathcal{O}\bigl((\log |G|+\log(1/δ))/\varepsilon^2\bigr)$ group-transformation-oracle queries. By comparison, standard group stochastic gradient descent (group-SGD), which samples a fresh transformation at every iteration, uses $\mathcal{O}(1/\varepsilon^4)$ transformation queries. Therefore, gradient descent with fixed sparse augmentation requires fewer transformation queries than both GD applied to the fully augmented objective and group-SGD. Our proof techniques, which may be of independent interest, establish a uniform approximation of the full group-averaged gradient field by a random group average using spectral properties of group-induced operators and tools from representation theory.

cs.LG↗

arXiv2026

The Complexity of Recognizing SDP Exactness for the Maximum Cut Problem

The standard semidefinite programming (SDP) relaxation of Max-Cut is exact when its optimum equals the maximum cut value. Delorme and Poljak resolved NP-completeness of recognizing exactness for weighted graphs and left the unweighted case open. We show that recognition is NP-complete even for connected simple unweighted graphs, and hence strongly NP-complete for nonnegative integer edge weights. The reduction provides an explicit SDP optimum and makes the additive integrality gap equal to the minimum number of unsatisfied clauses in the source formula. Recognition remains NP-complete even when an exact rational optimal primal--dual pair is supplied. We also establish strong NP-hardness of recognizing exactness of the Frieze--Jerrum Max-$k$-Cut relaxation for every fixed $k\ge3$, even for connected graphs with nonnegative integer edge weights. An independent bounded-weight construction gives a second proof for Max-Cut. Finally, reductions preserving the additive gap up to explicit factors establish strong NP-completeness of exactness recognition for a basic Max-DiCut SDP and NP-hardness for a Max-Bisection SDP.

math.OC↗

arXiv2026

A Fresh Look at Lamarckian Evolution and the Baldwin Effect

Baldwinian and Lamarckian evolution have existed for a long time in evolutionary algorithms (EAs) without ever dominating the academic literature or practical applications. In this work, we use modern empirical and theoretical methods to revisit Lamarckian and Baldwinian evolution and rigorously compare them with the generic Darwinian evolution. On the empirical side, we run a comprehensive suite of experiments on graphs from six different datasets from the recent GraphBench benchmark on Maximum Independent Set and Maximum Cut problems. Our results show that Baldwinian and Lamarckian evolution consistently outperform Darwinian evolution, confirming the great potential of local search augmented evolutionary algorithms. Notably, in the great majority of cases, all EAs outperform recent deep learning baselines and approach the performance of highly specialised heuristic and exact solvers. We furthermore report a high-performing set of generalist parameters for all studied evolution types that we hope will be of use to practitioners in future. On the theoretical side, we extend the existing DeceptiveLeadingBlocks benchmark to arbitrary block length $k$. For all constant $k$, we then prove asymptotically tight runtime bounds for the $(1+1)$ EA in the three evolution types on this benchmark. For Baldwinian evolution, these are independent of $k$, whereas for the other two evolution types, the runtimes steeply increase with growing value of $k$.

cs.NE↗

arXiv2026

Topology-Guided Modular Actor-Critic Learning for Continuous Systems under Temporal Objectives

We study formal policy synthesis for continuous-state stochastic systems under linear temporal logic specifications. The product of the system with the automaton of the specification has a hybrid state space with sparse rewards. We introduce a generalized optimal backup order, defined in reverse to a topological order over automaton states, that guides value backups and provably preserves optimality. We further present a model-free actor-critic algorithm whose policy evaluation solves a constrained optimization problem by the augmented Lagrangian method, yielding hyperparameter self-tuning, and prove its optimality and convergence in the tabular case. Since integer encodings of automaton states impose a spurious ordinal relationship on functions learned by one network, we dedicate a value and a policy network to each automaton state (modular learning). The algorithm matches or outperforms PPO, DQN, and A2C on CartPole, and on a Dubins car under a temporal specification the topological order and modular learning raise the success rate from 26.0% to 71.5%.

cs.AI↗

arXiv2026

Set-based state estimation of nonlinear discrete-time systems using constrained zonotopes and polyhedral relaxations

This paper presents a new algorithm for set-based state estimation of nonlinear discrete-time systems. A key step in such algorithms is to propagate a set (often a zonotope or constrained zonotope) through a nonlinear function. Existing methods accomplish this through conservative linearization procedures that are known to lead to severe overestimation in many cases. Here, we propose an alternative that avoids linearization using the so-called factorable representation of nonlinear functions. We use a recursive polyhedral relaxation technique based on this representation that is well-established in the global optimization literature but has not previously been used for set-based estimation. This technique is combined with constrained zonotope (CZ) technology to avoid the limitations of recursive computations with polyhedra in halfspace representation. The resulting state estimation method is fully automated, has attractive computational complexity (with one caveat discussed herein), and can provide significantly tighter enclosures than those resulting from linearization procedures in many cases. Numerical examples highlight the advantages of this approach relative to existing CZ methods based on the Mean Value Theorem and Difference of Convex functions (DC) programming.

eess.SY↗

arXiv2026

On the Effectiveness of the z-Transform Method in Quadratic Optimization

The z-transform of a sequence is a classical tool used within signal processing, control theory, computer science, and electrical engineering. It allows for studying sequences from their generating functions, with many operations that can be equivalently defined on the original sequence and its $z$-transform. In particular, the z-transform method focuses on asymptotic behaviors and allows the use of Taylor expansions. We present a sequence of results of increasing significance and difficulty for linear models and optimization algorithms, demonstrating the effectiveness and versatility of the z-transform method in deriving new asymptotic results. Starting from the simplest gradient descent iterations in an infinite-dimensional Hilbert space, we show how the spectral dimension characterizes the convergence behavior. We then extend the analysis to Nesterov acceleration, averaging techniques, and stochastic gradient descent.

cs.LG↗

arXiv2026

Accelerated Frank-Wolfe Algorithms: Complementarity Conditions and Sparsity

We develop new accelerated first-order algorithms in the Frank-Wolfe (FW) family for minimizing smooth convex functions over compact convex sets, with a focus on two prominent constraint classes: (1) polytopes and (2) matrix domains given by the spectrahedron and nuclear-norm balls. A key technical ingredient is a complementarity condition that captures solution sparsity---face dimension for polytopes and rank for matrices. We present two algorithms: (1) a purely linear optimization oracle (LOO) method for polytopes that has optimal worst-case first-order (FO) oracle complexity and, aside of a finite \emph{burn-in} phase and up to a logarithmic factor, has LOO complexity that scales with $r/\sqrtε$, where $ε$ is the target accuracy and $r$ is the solution sparsity (independently of the ambient dimension), and (2) a hybrid scheme that combines FW with a sparse projection oracle (e.g., low-rank SVDs for matrix domains with low-rank solutions), which also has optimal FO oracle complexity, and after a finite burn-in phase, only requires $O(1/\sqrtε)$ sparse projections and LOO calls (independently of both the ambient dimension and the sparsity level of optimal solutions). Our results close a gap on how to accelerate recent advancements in linearly-converging FW algorithms for strongly convex optimization, without paying the price of the dimension.

math.OC↗

arXiv2026

Eigenvalue stability and new perturbation bounds for the extremal eigenvalues of a matrix

Let $A$ be a full ranked $ n\times n$ matrix, with singular values $σ_1 (A) \ge \dots \ge σ_n (A) >0$. The condition number $κ(A):= σ_1(A)/σ_n(A)=\|A\|\cdot \|A\|^{-1}$ is a key parameter in the analysis of algorithms taking $A$ as input. In practice, matrices (representing real data) are often perturbed by noise. Technically speaking, the real input would be a noisy variant $\tilde A =A +E$ of $A$, where $E$ represents the noise. The condition number $κ(\tilde A)$ will be used instead of $κ(A)$. Thus, it is of importance to measure the impact of noise on the condition number. In this paper, we focus on the case when the noise is random. We introduce the notion of regional stability, via which we design a new framework to estimate the perturbation of the extremal singular values and the condition number of a matrix. Our framework allows us to bound the perturbation of singular values through the perturbation of singular spaces. We then bound the latter using a novel contour analysis argument, which, as a co-product, provides an improved version of the classical Davis-Kahan theorem in many settings. Our new estimates concerning the least singular value $σ_n(A)$ complement well-known results in this area, and are more favorable in the case when the ground matrix $A$ is large compared to the noise matrix $E$.

math.NA↗

arXiv2026

Ask Before You Optimize: Dynamic Pre-Formulation Clarification for Interactive Optimization

Large language models (LLMs) are increasingly used to formulate optimization models from natural-language problem descriptions, yet realistic operations research (OR) requests are often incomplete: missing objectives, constraints, or business rules can change the resulting mathematical program. Existing evaluations largely assume a complete specification and therefore overlook whether an agent knows when clarification is needed before modeling. We introduce OR-Clarify, a benchmark for pre-formulation clarification. Each task presents a partial public problem description, withholds structured hidden slots, and evaluates agents through bounded interaction with a simulated user. The benchmark supports both openended and choice-based clarification, and measures slot recovery, stopping behavior, silent assumptions, and interaction cost. We further propose Interactive Optimization (InterOPT), a two-stage framework that identifies unresolved formulation-critical gaps and uses them to guide whether to ask the next question or to stop. In our choice-based experiments, InterOPT substantially outperforms all baselines in exact slot recovery; in the open-ended setting, it remains competitive with strong prior methods. Together, OR-Clarify and InterOPT reframe OR assistance as a selective completeness decision: clarify when needed, stop when ready, and quantify what remains missing.

math.OC↗

arXiv2026

Hyperspectral Anomaly Detection via Group Sparse Low-Rank Tensor Factorization With Automatic Anomaly Grouping

Low-rank tensor modeling has become an effective tool for hyperspectral anomaly detection. However, existing methods still suffer from high computational cost and limited flexibility in characterizing spatially structured anomalies. To address these issues, this paper proposes a hyperspectral anomaly detection method based on group sparse low-rank tensor factorization with automatic anomaly grouping (GSAA). Specifically, the low tubal rank background is characterized by imposing group sparsity on tensor factors, which provides an efficient alternative to direct tensor rank regularization. For anomaly modeling, a latent grouping map is introduced to build an automatic anomaly grouping penalty, allowing anomaly groups to be adaptively inferred from the data rather than predefined at the pixel level. To further exploit complementary spectral and spatial information, GSAA is applied in both domains, and the resulting detection maps are fused to form a spectral--spatial version of GSAA, termed GSAA-SS. An efficient linearized alternating direction method of multipliers algorithm with convergence guarantee is developed to solve the resulting model. Experimental results on five real hyperspectral datasets demonstrate that the proposed method achieves superior detection performance and competitive computational efficiency compared with several state-of-the-art methods.

cs.CV↗