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math.OC: explore 64 source-linked works published from 2022 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 2026-09-14. Counts describe this index, not the complete source archives.

arXiv2026

Finite Sample Identification of Analytic Nonlinear Systems

This paper studies the identification of linearly parameterized nonlinear (LPN) systems. Although LPN systems share the same linear parameterization structure as linear systems, they are more challenging to identify. In particular, previous work has shown, through a counterexample based on a piecewise-affine system, that non-active exploration is generally insufficient for LPN system identification. In this paper, we consider LPN systems with real-analytic feature functions. We show that non-active exploration is sufficient for the identification of this class of systems by establishing non-asymptotic convergence rates of least-squares estimation and set-membership estimation. In addition, we provide counterexamples to show that non-active exploration may not be sufficient for system identification for non-real-analytic systems, even if those systems are infinitely differentiable. We present numerical experiments to further support and validate our theoretical results.

eess.SY↗
arXiv2026

On the suboptimality of stochastic MPC with varying constraint horizon

Enforcing stochastic state constraints over the full prediction horizon in Model Predictive Control (MPC) can be computationally demanding. Here we study stochastic MPC without terminal ingredients in which chance constraints are enforced only over a shorter constraint horizon. Using stochastic relaxed dynamic programming, we derive an explicit upper bound on the average expected closed-loop cost that depends on both prediction and constraint horizons. For linear quadratic problems with affine chance constraints and bounded uniform disturbances, we provide a deterministic reformulation via coordinate transformation and constraint tightening. Simulations illustrate the trade-off between computational effort and performance.

math.OC↗
arXiv2026

A note on the $Σ_2^P$-completeness of the Frobenius number

Given a finite set $A$ of natural numbers whose greatest common divisor is one, the Frobenius number $g(A)$ is the largest integer that is not a non-negative integer combination of the numbers in $A$. In a 2016 preprint, Matsubara states that given $A$ and $k$, deciding if $g(A) \geq k$ is $Σ_2^P$-complete. A decade has passed since without peer-reviewed publication of this result. At the same time, the community has found it difficult to verify this result. In this note, we give a write-up of the completeness proof based on Matsubara (2016).

cs.CC↗
arXiv2026

Equilibria in Network Constrained Markets with System Operator

We study a networked economic system composed of $n$ producers supplying a single homogeneous good to a number of geographically separated markets and of a centralized authority, called the market maker. Producers compete à la Cournot, by choosing the quantities of good to supply to each market they have access to in order to maximize their profit. Every market is characterized by its inverse demand functions returning the unit price of the considered good as a function of the total available quantity. Markets are interconnected by a dispatch network through which quantities of the considered good can flow within finite capacity constraints and possibly satisfying additional linear physical constraints. Such flows are determined by the action of a system operator, who aims at maximizing a designated welfare function. We model such competition as a strategic game with $n+1$ players: the producers and the system operator. For this game, we first establish the existence of pure-strategy Nash equilibria under standard concavity assumptions. We then identify sufficient conditions for the game to be exact potential with an essentially unique Nash equilibrium. Next, we present a general result that connects the optimal action of the system operator with the capacity constraints imposed on the network. For the commonly used Walrasian welfare, our finding proves a connection between capacity bottlenecks in the market network and the emergence of price differences between markets separated by saturated lines. This phenomenon is frequently observed in real-world scenarios, for instance in power networks. Finally, we validate the model with data from the Italian day-ahead electricity market.

cs.GT↗
arXiv2026

Aspiration-based Perturbed Learning Automata in Weakly-Acyclic Games with Noisy Utility Measurements

Reinforcement-based learning dynamics may exhibit several limitations when applied in a distributed setup. In (repeatedly-played) multi-player/action strategic-form games, and when each player applies an independent copy of the learning dynamics, convergence to (usually desirable) pure Nash equilibria cannot be guaranteed. Prior work has only focused on a small class of games, namely potential and coordination games. Furthermore, strong convergence guarantees (i.e., almost sure convergence or weak convergence) are mostly restricted to two-player games. To address this main limitation of reinforcement-based learning in repeatedly-played strategic-form games, this paper introduces a novel payoff-based learning scheme for distributed optimization in multi-player/action strategic-form games. We present an extension of perturbed learning automata (PLA), namely aspiration-based perturbed learning automata (APLA), in which each player's probability distribution for selecting actions is reinforced both by repeated selection and an aspiration factor that captures the player's satisfaction level. We provide a stochastic stability analysis of APLA in multi-player positive-utility weakly-acyclic games under the presence of noisy observations. We provide conditions under which convergence is attained (in weak sense) to the set of pure Nash equilibria. A methodology is also derived for calculating the stochastically stable equilibria through the aspiration action functional minimization, which simplifies the derivation of stochastically stable states. Conditions can then be derived for convergence to the Pareto efficient Nash equilibria. To the best of our knowledge, this is the first reinforcement-based learning scheme that provides global convergence guarantees in weakly-acyclic games and in a fully-distributed setup. A Monte-Carlo simulation study validates the derived conclusions.

cs.GT↗
arXiv2026

Tensor-Based Reduced-Order Modeling for Optimization-Based Inverse Problems

We develop a tensor reduced-order modeling (TROM) framework for optimization-based inverse problems governed by parameter-dependent dynamical systems. The approach approximates the parameter-to-observation map directly in tensor-train (TT) format using TT-SVD or TT-Cross and integrates it into a regularized nonlinear least-squares formulation. Beyond accelerating forward evaluations, the low-rank tensor structure reformulates the inverse problem in reduced coordinates, assembles Gauss--Newton quantities without forming the full observation-space Jacobian, and minimizes the TROM objective over the discrete parameter grid. This tensor optimization provides either a stand-alone approximate solution or a data-informed initialization for a subsequent Gauss--Newton solve. We study an inverse heat-transfer problem in a heterogeneous medium, where the parameters describe the locations and radii of low-conductivity inclusions, and a FitzHugh--Nagumo parameter-estimation problem with a highly nonconvex landscape. Numerical experiments assess reduced-order model error, measurement noise, regularization, initialization, spatial discretization, and increasing parameter dimension. The results show that TROM reproduces full-order inversion at substantially reduced online cost. They also demonstrate that reduced-coordinate inversion, tensor-based optimization, and appropriate regularization improve robustness in higher-dimensional, noisy, and strongly nonconvex regimes. For the continuous TROM inverse problem, we develop an error-to-inversion analysis. Under local strong convexity of the regularized FOM objective and parametric smoothness of the FOM observation map, the error between the parameters recovered with the full-order model and TROM is bounded by controlled uniform errors in the surrogate map and its Jacobian, together with local FOM and curvature quantities.

math.NA↗
arXiv2026

Fast Trainable Multilinear Bases for Image Compression

The Discrete Fourier Transform (DFT), the Discrete Cosine Transform (DCT), and their block-wise variants underpin most deployed image and video codecs. Their effectiveness rests on three properties: their runtime is near-linear (up to a polylogarithmic factor) in the image size, they are exactly invertible, and they carry few to no parameters. In this work, we generalize these bases to isometric multilinear bases, allowing a small number of extra parameters (polylogarithmic in the image size), while preserving all three properties. We develop a scheme to train a better transformation for a given image dataset: we use isometric tensor networks, inspired by quantum many-body theory, to parameterize the basis, and train it with Riemannian optimization. We show that training consistently improves performance, as our parameterized bases can represent the traditional DFT and DCT-IV (a variant of the DCT). Evidence is shown across natural photographs and line drawings. On Quick Draw line-drawing compression, for example, the best trained basis outperforms the block cosine transform used in the JPEG format by $20\%$ in terms of compressed data size.

eess.IV↗
arXiv2026

Column Number of Delta-modular matrices: Refined Analysis via Sauer Matrices

In this paper, we build upon the analysis initiated by Gennadiy Averkov and Matthias Schymura (2022) and establish that the number of distinct columns of a $Δ$-modular matrix $A \in \mathbb{Z}^{m \times n}$ of rank $m$ is $O(m^3 Δ)$. This upper bound was previously known only for odd values of $Δ$. Recall that a matrix is called $Δ$-modular if the maximum of the absolute values of its $m \times m$ minors equals $Δ$.

math.CO↗
arXiv2026

The Shadow Price of Intelligence: Quality Degradation in LLM Inference as a Supply Chain Problem

Large language model providers are compute constrained, and their universal response to congestion is to degrade service: route queries to smaller models, cut reasoning effort, truncate context. The industry's accounting says this saves money. We show the accounting is wrong, because it prices a query when the customer buys an answer. A degraded answer fails with some probability, and a failed answer either returns as a retry, inflating arrivals when the system is most loaded, or departs as churn, destroying lifetime value on a ledger no cost dashboard displays. We model inference allocation with three classical primitives: a newsvendor whose stockout cost is churned lifetime value, a geometric retry multiplier in which the recycled product is dissatisfaction, and a two-regime transient queue whose arrival rate is made endogenous by retries. Statically, there is a nonempty, measurable regime in which a cheaper model saves energy per satisfied answer while consuming strictly more capacity per satisfied answer, so the discount inverts exactly when capacity binds. Dynamically, a reactive throttle fired during a surge can cross an ignition threshold beyond which it manufactures more traffic than it sheds, and a release rule set below the degraded equilibrium converts a transient surge into a permanent degraded regime. With heterogeneous customers, throttling is a transportation problem in retry-inflated load whose optimal policy rations intelligence by critical ratio, class by class, and whose dual, the shadow price of intelligence, prices a marginal query by class and by hour; closed-form trajectories make it computable in milliseconds. Stochastic analysis sharpens rather than erodes the thesis: the ignition boundary acquires a predicted width, and noise punishes the reactive policy that parks the system against it. Under congestion, throttling is not a cost lever but a demand lever.

math.OC↗
arXiv2026

Sequential Euclidean connections with exponential memory: distributional performance and adversarial robustness

Points in the unit ball of $\mathbb R^d$ are processed sequentially. Each new point $p_i$ is connected to a state $x_{i-1}$ that summarizes earlier observations, after which $x_i=γx_{i-1}+(1-γ)p_i$, with $0\leqγ\leq1$. The cost is the sum of the $α$-powers of the connection lengths. This constant-gain rule interpolates between the input-order path and the star centered at the initial point. For independent uniform points, we establish the stationary insertion-length distribution and prove that it decreases in stochastic order as $γ$ increases. If $d+α>2$, or if $(d,α)=(1,1)$, the optimal constant parameter satisfies $1-γ_N^*=Θ(N^{-1/2})$, with an explicit asymptotic constant and closed bounds. For $α=1$, its leading expected tree length equals that of the center star and is eventually smaller than the expected lengths of both endpoint constructions. For $α=2$, the optimizer is unique and characterized exactly. For the same $N$, choosing $1-γ_N$ as a fixed positive multiple of $N^{-1/2}$ gives a sharp two-term expansion of the expected uniform-input cost and a maximal adversarial mean cost of $1+O(N^{-1/2})$. For every fixed $0\leqγ<1$ and $0<α\leq3$, the exact asymptotic adversarial value is $(2/(1+γ))^α$. When $d\geq2$, exponential weighting is within a factor smaller than $1.161^α$ of the best fixed nonnegative weighted rule with the same average look-back, for $0<α\leq3$. Comparison with the running mean highlights its time-homogeneous update, stationary coefficient profile, and fixed effective memory.

cs.CG↗
arXiv2026

B$^3$-PWL: GPU-Batched Branch-and-Bound for Piecewise-Linear Optimization with SOS2 Constraints

Piecewise-linear (PWL) optimization problems arise in many mixed-integer programming (MIP) optimization applications, including portfolio optimization, workforce scheduling, and resource allocation. But solving them to global optimality remains computationally expensive because branch-and-bound repeatedly solves LP relaxation subproblems. Existing solvers are largely CPU-centric, leaving the scalability of modern GPUs underutilized. Few prior GPU-accelerated branch-and-bound either targets neural network which is not suitable for general PWL optimization, or accelerates only auxiliary subroutines such as strong branching heuristics within CPU-centric MIP solvers. To bridge this gap, we propose B$^3$-PWL, a GPU-centric batched branch-and-bound framework for piecewise-linear optimization with Special Ordered Set of type 2 (SOS2) constraints. Our method solves batches of LP relaxation subproblems concurrently on the GPU using a first-order primal-dual solver, enabled by a specialized batched block-tiled sparse matrix kernel. To complement bound computation, we further introduce a unified feasibility search module that combines an SOS2 repair primal heuristic with a batched feasibility pump to rapidly obtain feasible incumbents and improve pruning efficiency. On a benchmark of 43 PWL-MIP instances, B$^3$-PWL achieves a 9.25x geometric-mean speedup over NVIDIA cuOpt while reaching high-quality feasible incumbents on every tested instance. On a public valve-point unit-commitment benchmark, it further outperforms NVIDIA cuOpt and the open-source CPU solvers SCIP and HiGHS, demonstrating the potential of first-order LP methods as the central engine of GPU-accelerated branch-and-bound.

math.OC↗
arXiv2026

Sharp Restricted Isometry Thresholds for Global Minima of Rank-Restricted Matrix LASSO

We determine the sharp restricted isometry threshold for recovery at global minima of the rank-restricted matrix LASSO. For target rank $r_{\star}$, if the rank-$k$ RIP constant satisfies $δ<δ_{\mathrm{sharp}}(k/r_{\star})$, where $δ_{\mathrm{sharp}}(t)=t/(4-t)$ for $0<t<4/3$ and $δ_{\mathrm{sharp}}(t)=\sqrt{(t-1)/t}$ for $t\ge4/3$, then every global minimizer has Frobenius error $\lesssim\sqrt{r_{\star}}λ$ for all $λ\gtrsim\|\mathcal{A}^{*}(ξ)\|_{\mathrm{op}}$ and at every search rank $r\ge r_{\star}$. The constants depend only on the RIP constant and $t=k/r_{\star}$, and in particular are independent of the search rank. When the rank restriction is inactive, the result specializes to the ordinary convex matrix LASSO. We also obtain the analogous results for sparsity-restricted vector LASSO. Conversely, we show that the threshold $δ<δ_{\mathrm{sharp}}(k/r_{\star})$ cannot be improved, due to the existence of counterexamples whose global minimizers fail to recover the ground truth.

stat.ML↗
arXiv2026

Bellman Search in Arbitrary Finite Dimension: A Self-Similar Cell Theorem and Effective Computability of Planar Shoreline Search

A shoreline-search path starts at the origin and must meet an unknown affine line, without knowing either its normal or its distance. We first establish a self-similar reduction theorem for homogeneous search problems whose historical information is a record profile updated by pointwise maximum. Two quasi-returns of the normalized state delimit a block that renews the required profile by itself; a short connector closes this block into a cell. Every finite-ratio path can therefore be approximated, with arbitrarily small loss, by repetitions of a single cell at all scales. The main chain is then made effective. A finite coding of the state space computably bounds the scale factor and normalized length of a nearly optimal cell. For planar Shoreline search, the support function of the convex hull gives an exact cell functional. A one-sided polygonalization then reduces the problem to a computable number of vertices, after which quantifier elimination decides whether a polygonal cell exists below a rational threshold. It follows that the optimal deterministic planar Shoreline value $C_2^*$ is a computable real: for every rational $ε>0$, an algorithm terminates with a rational interval of width at most $ε$ containing $C_2^*$. Additional results---sliding memory, Bellman transitions, deadlines, geometric filters, and relative equilibria---are presented separately as a toolbox for certified computation and for the study of spiral rigidity; they are not used in the computability proof.

cs.CG↗
arXiv2026

Bellman--Shoreline Search in Arbitrary Dimension: Exponential Vector Oscillators, Active Memory, Precession, and Effective Computability

We study online search for an unknown affine hyperplane in $\mathbb{R}^D$, for arbitrary fixed finite dimension. Building on a companion self-similar cell reduction and support-function formulation, we ask how the mechanism changes as the normal space grows from $\mathbb{S}^0$ to $\mathbb{S}^{D-1}$. In $D=1$, alternation and productivity yield an equal-ripple principle and the exact stationary constant $9$. In $D=2$, the analogous relative equilibrium is a logarithmic spiral whose bottleneck chord imposes tangency and selects the pitch. For exponential orbits $Γ(σ)=e^{κσ}ω(σ)$, we develop log-directional geometry, exponentially discounted memory, gauges, and recursive hyperspherical parametrizations. Without a shape ansatz, the bottleneck admits a certificate supported by at most $D$ historical suppliers, and at globally worst phases the current point lies on the active face. Within regular chambers we derive exact variation, tangency, pitch, age, and, in $D=3$, delay-system identities. Odd-dimensional obstructions, antipodal subclasses, and harmonic towers provide constraints and explicit candidate families but are not claimed globally optimal. Finally, the N-COMP theorem shows that $C_D^*$ is a computable real for every fixed finite $D$ and that algebraic polygonal $\varepsilon$-optimal cells can in principle be synthesized. Numerical screening through $D=10$ is kept separate from the proved results.

cs.CG↗
arXiv2026

A Systematic Approach to Mechanism Design with Stochastic Dynamic Stability

We consider a resource allocation problem with strategic agents that have private stochastic satisfaction functions and local constraints. To achieve a global optimal solution, we propose an incentive mechanism that induces a game among the agents. For the payment function of the mechanism, we construct a family of quadratic functions using the linear matrix inequality (LMI) approach that implements the social welfare maximizing outcome on the unique Nash equilibrium (NE) of the induced game while ensuring budget balance and individual rationality. Moreover, we propose a decentralized variable sample-size proximal best-response (VS-PBR) algorithm with Krasnoselskij iteration where only aggregate information is available to the agents. The algorithm is dynamically stable, as it is proven to converge in the mean-square sense to the NE of the game. The efficiency of the mechanism is then investigated on the Sioux Falls City transportation network, where electric vehicle (EV) users jointly select their destination and route.

eess.SY↗
arXiv2026

Uniform Statistical Convergence of Empirical Sinkhorn Potentials with Exponential and Polynomial Dependence on the Regularization Parameter

We study the empirical Sinkhorn estimator of the entropic optimal transport potentials under the uniform loss. Since the potentials are only unique up to additive constants, we measure the error using the quotient supremum norm, defined as $d_\infty([u],[v]) = \inf_{a\in\mathbb{R}}\|u-v-a\|_\infty$. For a fixed regularization parameter $\varepsilon>0$, we establish a non-asymptotic statistical rate of $n^{-1/2}$. This is achieved by combining the Birkhoff-Hopf contraction theorem with entropy bounds on normalized kernel sections. However, the constant in this bound grows exponentially with $1/ε$. To improve this, we isolate geometric conditions under which the empirical estimator maintains the $n^{-1/2}$ rate but features polynomial dependence on $1/\varepsilon$. The key requirement is a polynomial residual-stability estimate for the population Sinkhorn map. We provide sufficient criteria for this, including a polynomial contraction property and a local inverse estimate. Furthermore, we introduce two rigorously verifiable model classes an $\varepsilon$-weak residual-interaction class obtained after separable centering and another based on connected tight-edge graphs for fixed discrete costs where the polynomial rate is guaranteed without relying on abstract resolvent assumptions. Finally, we establish matching minimax lower bounds demonstrating that the $\varepsilon n^{-1/2}$ rate cannot be uniformly improved in the bounded-interaction regime.

stat.ML↗
arXiv2026

Nechvile-Transformed Spacecraft Dynamics and Propellant Computation in the 3-Body Problem

The uncontrolled equations of motion in the Nechvile frame for the restricted three-body problem have been well-known since at least the 1960s. It would seem that adding an external force to these equations is quite trivial: simply add an external force per mass term to the acceleration equations. Here we show that the last statement is not true. In fact, we show that the additive generic external force must be multiplied by the inverse of $(1 + e \cosθ)^3$ where $e$ is the relative eccentricity of the primaries and $θ$ is the true anomaly of the rotating frame located at the barycenter. Furthermore, when this result is combined with the mass flow rate equation, it generates several surprising results due to the mismatch between the resulting quadratic term and the cubic term in the equations of motion. This leads to a corresponding modification of the rocket equation itself. A Birkhoff-theoretic solution to an illustrative cislunar space mission problem shows propellent savings of 80% with the use of the correct cost functional. The popular quadratic cost utilizes more than $2X$ the minimum propellant consumption.

math.OC↗
Compare source metadata on this page
WorkPublishedSource identifierSource
Finite Sample Identification of Analytic Nonlinear Systems2026-08-302608.29908arxiv
On the suboptimality of stochastic MPC with varying constraint horizon2026-08-302608.30017arxiv
A note on the $Σ_2^P$-completeness of the Frobenius number2026-08-302608.30036arxiv
Rates of Convergence in the Central Limit Theorem for Markov Chains, with an Application to TD Learning2026-08-292401.15719arxiv
Equilibria in Network Constrained Markets with System Operator2026-08-292501.00191arxiv
Aspiration-based Perturbed Learning Automata in Weakly-Acyclic Games with Noisy Utility Measurements2026-08-292511.18418arxiv
Tensor-Based Reduced-Order Modeling for Optimization-Based Inverse Problems2026-08-292607.12613arxiv
Fast Trainable Multilinear Bases for Image Compression2026-08-292608.00053arxiv
Column Number of Delta-modular matrices: Refined Analysis via Sauer Matrices2026-08-292608.06215arxiv
The Shadow Price of Intelligence: Quality Degradation in LLM Inference as a Supply Chain Problem2026-08-292608.23986arxiv
Sequential Euclidean connections with exponential memory: distributional performance and adversarial robustness2026-08-292608.25298arxiv
B$^3$-PWL: GPU-Batched Branch-and-Bound for Piecewise-Linear Optimization with SOS2 Constraints2026-08-292608.28988arxiv
Sharp Restricted Isometry Thresholds for Global Minima of Rank-Restricted Matrix LASSO2026-08-292608.29018arxiv
Bellman Search in Arbitrary Finite Dimension: A Self-Similar Cell Theorem and Effective Computability of Planar Shoreline Search2026-08-292608.29032arxiv
Bellman--Shoreline Search in Arbitrary Dimension: Exponential Vector Oscillators, Active Memory, Precession, and Effective Computability2026-08-292608.29060arxiv
A Systematic Approach to Mechanism Design with Stochastic Dynamic Stability2026-08-292608.29130arxiv
Uniform Statistical Convergence of Empirical Sinkhorn Potentials with Exponential and Polynomial Dependence on the Regularization Parameter2026-08-292608.29152arxiv
Nechvile-Transformed Spacecraft Dynamics and Propellant Computation in the 3-Body Problem2026-08-292608.29271arxiv

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