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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 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

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

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

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

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↗

arXiv2026

Reinforcement learning to choose optimizers

No single optimization method is uniformly best for all problems, and the most suitable optimizer choice can change during a run. Existing approaches that change optimizer during execution typically predetermine part of the strategy: the portfolio is restricted to one algorithm class, the switch occurs once at a fixed time, or the frequency of decisions is treated as a hyperparameter rather than a learned one. We introduce "Reinforcement Learning to Choose Optimizers", which formulates the optimization algorithm choice as a sequential decision-making problem. At each decision, a recurrent policy reads the current run state and decides both which optimizer should be used next and for how long. The portfolio includes both gradient-based and derivative-free optimizers, and each switch passes on the current best solution and a representative step size. A context proxy conditions a gating network over expert heads, and training employs a decoupled actor-critic whose return is expressed in the same empirical runtime distribution metric used at evaluation. Training tasks and portfolio are designed jointly so that no optimizer dominates. On unseen problems, the learned policy outperforms every portfolio optimizer at all but the smallest budgets, and it remains robust under distribution shift.

cs.NE↗

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 proved NP-completeness of recognizing exactness for weighted graphs and left the unweighted case open. We prove that recognizing exactness is NP-complete even for connected simple unweighted graphs, and hence strongly NP-complete for graphs with nonnegative integer weights. The reduction represents the clauses of a linear monotone NAE-4-SAT instance by edge-disjoint copies of $K_6$. An explicit vector assignment attains a common SDP bound, and the additive integrality gap equals the minimum number of unsatisfied clauses. Hardness persists for simple unweighted graphs when an exact rational optimal primal--dual pair is supplied. Shared simplex anchors establish strong NP-hardness of recognizing exactness of the Frieze--Jerrum Max-$k$-Cut SDP relaxation for every fixed $k\ge3$, even for connected graphs with nonnegative integer edge weights. We also give an independent bounded-weight sum-of-squares proof of strong NP-completeness of recognizing exactness of the Max-Cut SDP, together with gap-preserving reductions establishing strong NP-completeness of exactness recognition for a basic Max-DiCut SDP and NP-hardness for a Max-Bisection SDP.

math.OC↗

arXiv2026

Operational Regimes in Non-Convex Optimization: A Multiplier-Based Taxonomy

This paper introduces a structural taxonomy for constrained non-convex optimization based on the signature of Lagrange multipliers at KKT stationary points. Leveraging a unified game-theoretic interpretation of eight classical algorithm families--including block coordinate descent, ADMM, generalized Benders decomposition, successive convex approximation, interior-point methods, mirror descent, Frank-Wolfe, and Riemannian gradient descent--we show that the normalized multiplier vector carries an algorithm-independent structural fingerprint. Four scale-free shape features of this vector partition the dual space into five operational regimes: Unconstrained, Resource-Limited, Saturation, Strongly-Coupled, and Hybrid. We establish four structural theorems characterizing the partition: invariance under natural KKT symmetries, local stability under data perturbation with explicit Lipschitz margins from Robinson's strong regularity, codimension-one regime transitions, and the topological identification of the Hybrid regime as the Lebesgue-null boundary of the core regimes. A linear-time classifier is proposed with provable guarantees on correctness, iteration stabilization, sample complexity, and online tracking under data drift. Numerical experiments on 104 mixed-integer nonlinear programs and a downlink beamforming instance validate the theoretical predictions. The framework provides a foundational tool for regime-aware algorithm design and robustness analysis in non-convex optimization.

math.OC↗

arXiv2026

On the Exponential Circuit Imbalance of the Ben-Tal Nemirovski Approximation

Dadush et al.\ (2024) recently developed a scaling-invariant layered least squares algorithm for linear programming whose complexity depends on the optimal condition measure $\barχ_A^*$. Their work builds on Vavasis and Ye's (1996) algorithm whose running time depends only on the constraint matrix $A$ through the condition number $\barχ_A$. Monteiro-Tsuchiya (2003) defined the optimal condition number $\barχ_A^*$ as the maximum $\barχ_{AD}$ achievable over all positive diagonal column rescalings $D$. Dadush et al.\ (2024) introduced the optimal circuit imbalance measure $κ_W^*$, which serves as a lower bound for $\barχ^*_A$. Instances with artificially large optimal circuit imbalance measures $κ_W^*$ can be easily constructed; however, finding naturally occurring examples where this optimal scaling-invariant measure grows exponentially is of independent interest. In this paper, we show that the Ben-Tal Nemirovski (BN) linear programming approximation of the unit disk provides such an example. By explicitly constructing circuits in the kernel of the BN formulation, we prove that the optimal circuit imbalance measure $κ_W^*$ grows exponentially in the number of approximation steps. Since $κ_W^*$ lower bounds $\barχ_A^*$, our result demonstrates that the BN approximation yields an exponentially ill-conditioned family of constraint matrices.

cs.CC↗

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

SS-ESOAP: Self-Scaled Adaptive Preconditioning for Physics-Informed Learning

Physics-informed neural networks (PINNs) often face ill-conditioned objectives that limit high-accuracy training. Dense quasi-Newton methods improve local conditioning but require expensive optimizer state, while Kronecker-factored methods such as SOAP scale to larger networks but rely on periodic basis updates. We introduce \method, which augments SOAP-style preconditioning with a scalar secant-energy correction adapted to Kronecker geometry and an adaptive basis update followed by variance-state downscaling. We characterize the directional secant matching induced by the scalar correction and give a bound on variance-state mismatch across basis changes. Across eight PDE benchmarks, \method attains the lowest final residual on six, including Burgers and Boussinesq, while SOAP-family baselines perform better on Gray-Scott and Ginzburg-Landau. On Boussinesq, \method reaches a residual of $10^{-5}$ in 4.1 hours with 9.2 GB peak VRAM, while Adam does not reach this target within 14 hours. Three-seed $L^2$ and $H^1$ errors on four representative PDEs support the link between lower residuals and improved solution accuracy. These results position \method as a scalable option for stiff, high-accuracy physics-informed training, rather than a uniform replacement for existing optimizers.

cs.LG↗

arXiv2026

Deterministic Non-Smooth Safety via Dual-Algebraic Control Barrier Functions

This paper presents a dual-algebraic framework for control barrier functions (CBFs) that guarantees deterministic execution using exclusively elementary arithmetic. We develop this deterministic approach to solve a fundamental bottleneck in safety-critical control: pointwise minima compose intersecting safe sets, but generate non-smooth boundaries where standard Lie derivatives fail. Existing mathematical workarounds inject approximation bias, probabilistic non-determinism, or combinatorial execution delays that impede hard real-time hardware certification. By embedding the system state and vector field into the dual-number ring, our method extracts both the composite barrier value and its exact directional derivative in a single evaluation. The standard floating-point minimum deterministically isolates a single vertex of the Clarke generalized gradient for the quadratic-program solver. We prove this selected vertex constitutes a Clarke subgradient and the resulting simultaneous-enforcement safety filter guarantees forward invariance. The arithmetic overhead remains a fixed constant factor, independent of state dimension and constraint count. We extend this framework to finite $\min$/$\max$ Boolean compositions, for which enforcement of the routed constraint of each $δ$-active clause guarantees forward invariance, and to systems of higher relative degree, for which a bivariate truncated-dual evaluation extracts the control coupling without symbolic differentiation. Three numerical examples illustrate the computational scaling.

math.OC↗

arXiv2026

LMI Properties and Applications in Systems, Stability, and Control Theory

Linear matrix inequalities (LMIs) commonly appear in systems, stability, and control applications. Many analysis and synthesis problems in these areas can be solved as feasibility or optimization problems subject to LMI constraints. Although most well-known LMI properties and manipulation tricks, such as the Schur complement and the congruence transformation, can be found in standard references, many useful LMI properties are scattered throughout the literature. The purpose of this document is to collect and organize properties, tricks, and applications related to LMIs from a number of references together in a single document. In this sense, the document can be thought of as an "LMI encyclopedia" or "LMI cookbook." Proofs of the properties presented in this document are not included when they can be found in the cited references in the interest of brevity. Illustrative examples are included whenever necessary to fully explain a certain property. Multiple equivalent forms of LMIs are often presented to give the reader a choice of which form may be best suited for a particular problem at hand. The equivalency of some of the LMIs in this document may be straightforward to more experienced readers, but the authors believe that some readers may benefit from the presentation of multiple equivalent LMIs.

eess.SY↗

arXiv2026

Reduced order model for parametric Boltzmann equation and its application to inverse problems

The Boltzmann equation plays an important role in modeling mesoscopic behavior in a wide range of scientific and engineering applications. However, its numerical solution is computationally expensive due to the high dimensionality of the model and the nonlinear nonlocal collision operator, especially for steady-state problems that require iterative solvers. This cost becomes prohibitive for inverse problems, where the induced optimization problem requires repeated forward solves. In this work, we propose a reduced-order model (ROM) for the parametric Boltzmann equation to address this computational challenge. The ROM constructs a low-dimensional approximation space for the parameter-induced solution manifold through a residual-based greedy strategy, and the reduced solution is then obtained via residual minimization over the reduced space, subject to mass conservation. The overall efficiency of the ROM is achieved by exploiting the quadratic structure of the collision operator and a precomputed separable approximation of the collision kernel. The resulting ROM is further applied to a thermally-driven inverse problem for reconstructing collision parameters from the observed macroscopic temperature data. This is accomplished either by directly replacing the PDE constraint with the ROM, leading to a bilevel optimization formulation, or by reformulating the task as a single-level optimization problem through the Karush--Kuhn--Tucker (KKT) conditions. Numerical experiments in both collision-dominated and transport-dominated cases are performed to demonstrate the efficiency and accuracy of the proposed ROM and its effectiveness in inverse problems. In particular, the resulting inverse problem is computationally much more tractable, achieving speedups of several orders of magnitude over that based on the full-order model while maintaining comparable accuracy.

math.NA↗

arXiv2026

Near-Optimal Mechanisms for Resource Allocation Without Monetary Transfers

We study the problem in which a central planner sequentially allocates a single resource to multiple strategic agents using their utility reports at each round, but without using any monetary transfers. We consider general agent utility distributions and two standard settings: a finite horizon $T$ and an infinite horizon with $γ$ discounts. We provide general tools to characterize the convergence rate between the optimal mechanism for the central planner and the first-best allocation if true agent utilities were available. This heavily depends on the utility distributions, yielding rates anywhere between $1/\sqrt T$ and $1/T$ for the finite-horizon setting, and rates faster than $\sqrt{1-γ}$, including exponential rates for the infinite-horizon setting as agents are more patient $γ\to 1$. On the algorithmic side, we design mechanisms based on the promised-utility framework to achieve these rates and leverage structure on the utility distributions. Intuitively, the more flexibility the central planner has to reward or penalize any agent while incurring little social welfare cost, the faster the convergence rate. In particular, discrete utility distributions typically yield the slower rates $1/\sqrt T$ and $\sqrt{1-γ}$, while smooth distributions with density typically yield faster rates $1/T$ (up to logarithmic factors) and $1-γ$.

cs.GT↗

arXiv2026

Efficient Hessian-Free Methods for Multi-Objective Bilevel Optimization with Nonconvex Lower Level

Multi-objective bilevel optimization has wide applications in the AI area such as automated learning and multi-task meta-learning. Although recently some works have been begun to study the multi-objective bilevel optimization, the proposed methods rely on the (strongly) convex lower level problems. In fact, these multi-objective bilevel learning problems are generally nonconvex, and particularly their lower level problems are nonconvex. To fill this gap, we propose a class of Multi-Objective Moreau Envelope based Hessian-free Algorithms (MOMEHA) for the multi-objective bilevel learning problems with nonconvex lower level. Specifically, our method uses the Moreau envelope to relax the original problem into a multi-objective single-level optimization with an envelope constraint. In particular, our method retains computational advantages of being single-loop and Hessian-free in the multi-objective setting by incorporating a smooth weighted Tchebycheff scalarization. Furthermore, we propose a momentum-based variant of MOMEHA (i.e., MB-MOMEHA) method for the stochastic multi-objective bilevel learning problems. In theory, we provide the convergence properties of our algorithms under both deterministic and stochastic setting. Some experiments on few-shot meta-learning and neural architecture search demonstrate that our methods outperform the existing approaches in Pareto front, validating its effectiveness and robustness.

math.OC↗

arXiv2026

Assessing Autonomous Mobility-on-Demand Services and the Impacts of Operational Strategies: A Case Study of Chengdu, China

The Autonomous Mobility-on-Demand (AMoD) service is emerging as a potential alternative to on-demand urban mobility, but its operational performance relative to traditional street-hailing services and the effectiveness of related operational strategies remain unclear. This study presents a simulation framework integrating a graph theory-based trip-vehicle matching mechanism and uses historical street-hailing operations data to simulate AMoD services in Chengdu, China. The operational performance of these two urban mobility modes is evaluated using three key performance indicators: average passenger waiting time (APWT), average deadheading mileage (ADM), and average deadheading energy consumption (ADEC). We further evaluate the impacts of four operational strategies on simulated AMoD performance: vehicle repositioning, fleet size management, geofencing, and request rejection. Simulation results indicate that, under the same historical trip demand, fleet-size constraints, and road network as the observed street-hailing system, the simulated AMoD service is estimated to have lower values of APWT, ADM, and ADEC by 73.3% to 83.4%, 75.0%, and 74.0%, respectively, reflecting the potential operational gains associated with centralized dispatch in simulation settings. These differences are most pronounced during early-morning low-demand hours and in remote areas such as airports.

math.OC↗