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arXiv2026

Flow Shop Scheduling with Stochastic Reentry

We study flow shop scheduling with stochastic reentry, where jobs must complete multiple passes through the entire shop, and the number of passes that a job requires for completion is drawn from a discrete probability distribution. The goal is to find policies that minimize performance measures in expectation. Our main contribution is a reduction to a stochastic scheduling problem on identical parallel machines augmented by machine arrivals. This reduction preserves objective values and enables the transfer of structural results and performance guarantees from the auxiliary problems to the reentrant flow shop setting. We demonstrate the usefulness of this reduction by proving the optimality of simple priority policies for minimizing the makespan and the total completion time in expectation under geometric and, more generally, monotone hazard rate distributions. For minimizing the total weighted completion time, we derive an approximation guarantee for a simple priority policy that depends only on the squared coefficient of variation of the underlying distributions. Our results constitute the first optimality and approximation guarantees for flow shops with stochastic reentry and demonstrate that established scheduling policies naturally extend to this setting through the proposed reduction.

cs.DS↗

arXiv2026

Adaptively Incorporating Directional Hints into Zeroth-Order Optimization

We study zeroth-order optimization of non-convex functions with the aid of directional hints, which are cheap but potentially inaccurate approximations of the true gradient direction, given by linear subspaces at each iteration. To leverage these hints adaptively while maintaining robustness to their quality, we introduce Control-Variate Zeroth-Order Descent (CV-ZOD), a new framework that refines the classical zeroth-order gradient estimator with a control variate that can be set based on the directional hints. We first show that the oracle algorithm that optimally sets the reference vector and step size at each iteration achieves a convergence rate that interpolates between the first-order $O(1/T)$ rate and the zeroth-order $O(d/T)$ rate, depending on the quality of the hints along the trajectory. We then develop a practical variant of CV-ZOD that achieves the same oracle guarantee up to logarithmic factors, without any prior knowledge of the hint quality. We validate the method empirically on simulation-based scientific optimization tasks, demonstrating sustained progress on non-convex landscapes where zeroth-order descent is slower and existing guided methods stall as guidance deteriorates.

cs.LG↗

arXiv2026

Observability conditions for neural state-space models with eigenvalues and their roots of unity

We operate through the lens of ordinary differential equations and control theory to study the concept of observability in the context of neural state-space models and the Mamba architecture. We develop strategies to enforce observability, which are tailored to a learning context, specifically where the hidden states are learnable at initial time, in conjunction to over its continuum, and high-dimensional. We also highlight our methods emphasize eigenvalues, roots of unity, or both. Our methods effectuate computational efficiency when enforcing observability, sometimes at great scale. We formulate observability conditions in machine learning based on classical control theory and discuss their computational complexity. Our nontrivial results are fivefold. We discuss observability through the use of permutations in neural applications with learnable matrices without high precision. We present two results built upon the Fourier transform that effect observability with high probability up to the randomness in the learning. These results are worked with the interplay of representations in Fourier space and their eigenstructure, nonlinear mappings, and the observability matrix. We present a result for Mamba that is similar to a Hautus-type condition, but instead employs an argument using a Vandermonde matrix instead of eigenvectors. Our final result is a shared-parameter construction of the Mamba system, which is computationally efficient in high exponentiation. We develop a training algorithm with this coupling, showing it satisfies a Robbins-Monro condition under certain orthogonality, while a more classical training procedure fails to satisfy a contraction with high Lipschitz constant.

cs.LG↗

arXiv2026

Solution Methods for Infinite-Dimensional Generalized Disjunctive Programming

Generalized disjunctive programming (GDP) expresses mixed discrete-continuous decisions through Boolean indicators and disjunctions, and can be systematically solved via a library of methods proposed in the literature. The recent InfiniteGDP abstraction lifts this modeling layer to continuous domains such as time, space, and uncertainty, but only the big-M and hull reformulations, the two endpoints of the relaxation spectrum, have been generalized to the infinite setting. This work closes this gap by generalizing four other GDP solution methods to infinite-dimensional optimization: the multiple big-M reformulation, P-split reformulation, cutting plane reformulation, and the logic-based outer approximation algorithm. It further proposes MBM-GP, a novel Gaussian-process variant of multiple big-M that learns the big-M function over the infinite domain from a small subset of the subproblem solves. Moreover, these approaches are implemented in the Julia package InfiniteDisjunctiveProgramming.jl. The methods are benchmarked on case studies arising in dynamic and stochastic optimization. The results demonstrate how the generalized solution methods can outperform big-M and hull, with MBM-GP retaining the tightness of multiple big-M at a fraction of its reformulation cost.

math.OC↗

arXiv2026

Approximating CDTW Distance of Piecewise Algebraic Curves

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

cs.CG↗

arXiv2026

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

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

cs.CV↗

arXiv2026

Distributed Linear Programming on GPU Clusters at Extreme Scale

Large linear programs can exceed the memory of a single compute node. Although first-order methods replace sparse factorizations with GPU-suited matrix-vector products, other solver phases can reintroduce a single-node memory limit. We present SHARDLP, a distributed GPU LP solver that keeps the matrix and primal-dual state partitioned from sharded input through solution output. On the Google PDLP benchmark, SHARDLP reaches the published criterion on nine of eleven instances, compared with eight in the published CPU PDLP study. On the largest benchmark, eight H200 GPUs solve a 1.185-billion-variable, 6.338-billion-nonzero LP in 9.9 minutes; the published CPU experiment reports 21.06 hours on different hardware. Beyond this benchmark, separately checked multi-node solves reach up to 13.604 billion variables and 40.807 billion nonzeros, while validated executions span up to 76 GPUs across 29 compute nodes. For column-partitioned solves, support-aware communication skips GPUs that store no coefficients for a row; on an LP with 2.76 billion nonzeros, it cuts modelled communication by 92.97% and improves solver time by 1.27x-1.52x

math.OC↗

arXiv2026

Accelerating Fourier--Motzkin elimination: redundancy removal and the choice of variable elimination order

Fourier-Motzkin elimination computes an inequality description of the projection of a polyhedron onto a subset of its coordinates by eliminating one variable at a time. It is used in several areas of optimisation and computer science, and it is a standard way of obtaining the entropic constraints of a causal structure, where the marginalisation over the latent variables produces such a projection. Its limitation is the growth of the intermediate systems of inequalities, which can be doubly exponential in the number of eliminated variables even though the projection itself grows only as a single exponential. In practice the computational overload of the method therefore depends on two choices: how the redundant inequalities are removed after each step, and the order in which the variables are eliminated. We consider both. We first show, by an explicit example, that Imbert's redundancy test cannot be interleaved with redundancy removal by linear programming. We show that the two methods, however, can be combined soundly if the derivation records used by Imbert's test are re-initialised after every step at which linear programming is used. We then propose a rule for choosing the elimination order of the variables that gives a significant computational advantage, however, at the cost of increased resource usage. We demonstrate this advantage on some random polytopes, where the rule reduces the running time by factors of between 6 and 25 compared with the same elimination under a fixed order. For entropic descriptions of causal structures, with more than 250 inequalities and more than 100 variables to eliminate, our rule keeps the number of inequalities handled at each step one to two orders of magnitude lower than a fixed order.

cs.CC↗

arXiv2026

Learning-enabled Acceleration of Scenario-based Model Predictive Control

Scenario-based model predictive control (SBMPC) is a variant of model predictive control (MPC) that explicitly accounts for uncertainty by optimizing control actions over multiple predicted scenarios. However, its computational complexity increases rapidly with the number of scenarios and prediction horizon, limiting its applicability to real-time planning and control. This paper presents a learning-accelerated Alternating Direction Method of Multipliers (ADMM) algorithm for efficiently solving SBMPC problems by leveraging parallel computing and Moreau envelope learning, while maintaining high solution accuracy. We reformulate the SBMPC problems into consensus forms that can be decomposed via ADMM, separating the scenario-dependent dynamics from non-anticipativity constraints and enabling parallel updates across scenarios and time steps. Building on this decomposition, we utilize a learning-to-optimize scheme that leverages Moreau envelope learning of the cost function to accelerate the primal update in ADMM, thereby reducing computation time. The proposed framework is evaluated on a microgrid energy management problem subject to load and renewable generation uncertainties. Comparisons with IPOPT and MadNLP, two popular and modern nonlinear programming solvers, demonstrate substantial computational speedups while maintaining reliable closed-loop control performance.

math.OC↗

arXiv2026

Anchored Regularized Direct Least Squares (ARDLS): Integrating Established Prioritization Operators for Priority Elicitation in the Analytic Hierarchy Process

Pairwise reciprocal matrices are fundamental to the Analytic Hierarchy Process (AHP),a decision-making model. While the Direct Least Squares (DLS) method provides an intuitive mechanism for deriving priority vectors without complex transformations, it is susceptible to solution non-uniqueness. Under high levels of inconsistency, such as severe cyclic contradictions, the DLS optimization landscape becomes non-convex, yielding multiple distinct global minima. Consequently, priority rankings become unstable and critically dependent on initial algorithmic guesses. Furthermore, established prioritization operators (POs), including normalization techniques, the Eigenvector method, Singular Value Decomposition, Cosine Maximization, and the Pseudo-Inverse Gram Matrix (the closed-form solution of Weighted Least Squares), frequently generate disparate outcomes. To overcome these structural deficiencies, this paper introduces the Anchored Regularized Direct Least Squares (ARDLS) optimization model as a harmonizing framework. ARDLS integrates uniquely determined established POs as theoretical anchors within a regularization penalty. This integration systematically breaks mathematical symmetries and tilts the optimization landscape to guarantee convergence upon a single, unique global minimum. By minimizing the root mean square variance (RMSV) of the initial baseline vectors, ARDLS effectively unifies these divergent solutions. Comprehensive numerical experiments validate that the framework successfully fine-tune the solution of established POs by reducing RMSV while ensuring strict mathematical uniqueness. The practical utility of the method is further demonstrated through a numerical case study resolving an innovation fund dilemma in FinTech project selection. The proposed ARDLS approach offers a robust alternative to classical AHP across a wide range of decision-making domains.

math.OC↗

arXiv2026

Bilevel MPC for Linear Systems: A Tractable Reduction and Continuous Connection to Hierarchical MPC

Model predictive control (MPC) has been widely used in many fields, often in hierarchical architectures that combine controllers and decision-making layers at different levels. However, when such architectures are cast as bilevel optimization problems, standard KKT-based reformulations often introduce nonconvex and potentially nonsmooth structures that are undesirable for real-time verifiable control. In this paper, we study a bilevel MPC architecture composed of (i) an upper layer that selects the reference sequence and (ii) a lower-level linear MPC that tracks such reference sequence. We propose a smooth single-level reduction that does not degrade performance under a verifiable block-matrix nonsingularity condition. In addition, when the problem is convex, its solution is unique and equivalent to a corresponding centralized MPC, enabling the inheritance of closed-loop properties. We further show that bilevel MPC is a natural extension of standard hierarchical MPC, and introduce an interpolation framework that continuously connects the two via move-blocking. This framework reveals optimal-value ordering among the resulting formulations and provides inexpensive a posteriori degradation certificates, thereby enabling a principled performance-computational efficiency trade-off.

eess.SY↗

arXiv2026

Truth Revelation, Information Hiding, or Misinformation: Characterization of Equilibrium Outcomes in Signaling Games

In signaling games where a sender and a receiver have misaligned criteria, equilibrium behavior may lead to fully revealing, quantized, or randomized policies. Notably, the first arises in statistical decision theory and classical communication theoretic problems involving a fully aligned sensor and receiver, the second arises in Nash theoretic simultaneous signaling games, and the last may appear in Stackelberg type (leader-follower) Bayesian signaling games. In this paper, we investigate the Bayesian persuasion problem involving a receiver that tries to estimate the source. We show that for certain payoff structures, the equilibrium solution is such that a source observation is mapped to distinct messages with nonzero probabilities. More specifically, we completely characterize conditions under which the sender requires randomization for the Bayesian persuasion problem involving general sources with finite cardinality. In particular, regardless of whether the equilibrium solution under a deterministic policy restriction is fully revealing, quantized or noninformative, there exists a randomized sender policy that improves the sender's payoff under certain conditions characterized in the paper. Moreover, we provide an algorithmic procedure to obtain the Bayesian persuasion solution, where the algorithm compares the payoffs with finitely many posterior probability combinations. We also consider fully aligned and completely misaligned payoff structures, where the solutions respectively involve a fully revealing sender and a noninformative sender. Then, we unify these results by proving that if the sender's expected payoff with respect to posterior distributions is continuous, then the equilibrium solution involves either a fully revealing sender or a noninformative sender.

cs.GT↗

arXiv2026

Generalized Regret Analysis of Thompson Sampling using Fractional Posteriors

Thompson sampling (TS) is one of the most popular and earliest algorithms to solve stochastic multi-armed bandit problems. We consider a variant of TS, named $α$-TS, where we use a fractional or $α$-posterior ($α\in(0,1)$) instead of the standard posterior distribution. To compute an $α$-posterior, the likelihood in the definition of the standard posterior is tempered with a factor $α$. For $α$-TS we obtain both instance-dependent $\mathcal{O}\left(\sum_{k \neq i^*} Δ_k\left(\frac{\log(T)}{C(α)Δ_k^2} + \frac{1}{2} \right)\right)$ and instance-independent $\mathcal{O}(\sqrt{KT\log K})$ frequentist regret bounds under very mild conditions on the prior and reward distributions, where $Δ_k$ is the gap between the true mean rewards of the $k^{th}$ and the best arms, and $C(α)$ is a known constant. Both the sub-Gaussian and exponential family models satisfy our general conditions on the reward distribution. Our conditions on the prior distribution can be easily satisfied by a density that is positive, continuous, and bounded. We also establish another instance-dependent regret upper bound that matches (up to constants) to that of improved UCB [Auer and Ortner, 2010]. Our regret analysis carefully adapts and combines recent theoretical developments in the non-asymptotic concentration analysis and Bernstein-von Mises type results for the $α$-posterior distribution. Moreover, our analysis does not require additional structural properties such as closed-form posteriors or conjugate priors.

stat.ML↗

arXiv2026

Support Discovery With Iteratively Reweighted Least Squares for Fixed-Charge Network Flow

The fixed-charge network flow problem (FCNFP) couples continuous flow allocation with discrete arc-activation decisions, making it a canonical but computationally challenging model for a variety of network design and resource allocation problems. Exact mixed-integer linear programming formulations capture the fixed-charge structure faithfully, but often become difficult to solve on large networks. We propose a scalable continuous-optimization algorithm for large-scale single-commodity FCNFP based on an iteratively reweighted least-squares (IRLS) framework. The method replaces the discontinuous fixed-charge and linear arc cost objective with a smooth nonconvex Lasry--Lions surrogate and solves a sequence of weighted quadratic flow subproblems. Each subproblem is solved by a warm-started dual semismooth Newton method whose Newton systems have weighted graph-Laplacian structure, enabling the use of modern Laplacian solvers. To further improve the discovered arc supports of the challenging underlying combinatorial problem, we also develop an algorithmic variant that incorporates objective-driven perturbation restarts and an anchor-union restricted search that jointly leverages supports discovered by IRLS and by complementary FCNFP heuristics. Computational experiments on 410 benchmark, synthetic, and large-scale instances show that our method obtains the best objective quality among the evaluated scalable FCNFP algorithms, with a mean gap of $1.316\%$ to a time-limited MILP reference and a win-or-tie rate of $90.0\%$ among the non-MILP methods. The results indicate that combining smooth continuous optimization with support-level search is an effective strategy for producing high-quality feasible solutions to large-scale FCNFP.

math.OC↗

arXiv2026

Parameterized Complexity of Scheduling Problems in Robotic Process Automation

This paper studies the growing domain of Robotic Process Automation (RPA) problems. Motivated by scheduling problems arising in RPA, we study the parameterized complexity of the single-machine problem with precedence constraints, release times, and deadlines (i.e., the problem known as $1|\operatorname{prec},r_j,d_j|*$ in the three-field notation). We focus on parameters naturally linked to RPA systems, including chain-like precedences, the number of distinct processing times, and the structure of the time windows. We show that the problem is strongly XNLP-hard parameterized by the number of chains, even with only two prescribed processing times and two distinct time-window lengths. The problem remains XNLP-hard even under prec-consistent time windows. On the positive side, we obtain polynomial-time algorithm when all jobs share a single time-window length and FPT when the processing times, release times and deadlines are chain-uniform. We also show that the problem lies in XNLP when parameterized by the width of the precedence relation either when the instance is encoded in unary or the processing times are bounded.

cs.DS↗

arXiv2026

Recovery Theory for Projected Power Iterations in Permutation Synchronization

We study the projected power method (PPM) for synchronizing $n$ unknown permutations of $m$ objects under a possibly sparse uniform corruption model. Each pair is observed with probability $p$, and an observed measurement is uncorrupted with probability $π_0$ and is otherwise an independent uniform permutation. Under $\log m=o(npπ_0^2)$, we prove exact one-step recovery (with high probability) of each prescribed block for an independent estimate with a fixed positive majority of correct blocks. When $np\ge C_0\log n$ and $m=o(npπ_0^2)$, we prove that one high-probability event yields a block-error contraction simultaneously for every estimate whose optimally aligned error is at most $0.5-ε$. The contraction factor is $O(m/(npπ_0^2))$ and the error floor is $O(e^{-cnpπ_0}+e^{-cnpπ_0^2}+{\log n}/{n})$. Consequently, one update maps every possibly data-dependent estimate in this basin to vanishing block error, and all subsequent iterates remain almost exact uniformly over the iteration index. The one-step and trajectory results extend to independent, non-identically distributed, permutation-valued corruptions with mean $m^{-1} \mathbf{1}\mathbf{1}^{\top}$. Under the uniform model, a reference-block spectral initializer has aligned block error $O_{\mathbb P}(m/(npπ_0^2))$, yielding an end-to-end almost-exact recovery guarantee. Under a stronger all-block signal condition, PPM reaches exact recovery after finitely many iterations. The theory transfers exactly to partial permutations with common support; for varying supports, we establish deterministic and probabilistic co-visibility margins.

stat.ML↗

arXiv2026

CP-WSP: A Declarative CP-SAT Framework for Configurable Multi-Constraint Workforce Scheduling

Workforce scheduling is an NP-hard combinatorial optimization problem requiring simultaneous satisfaction of labor regulations, coverage requirements, employee preferences and operational objectives. Existing CP formulations typically model simplified instances with 6-12 constraints at shift-level granularity and critically lack explicit support for: mandatory break scheduling with midpoint placement control; acuity weighted workload equity; sub-shift temporal granularity enabling demand-driven staffing; inter-week schedule stability; and cross-midnight shift patterns common in 24-hour operations. This paper presents CP-WSP: a declarative CP-SAT framework enforcing 14 hard constraints as mathematically inviolable requirements (zero regulatory violations by construction) while optimizing 15 soft objectives through a unified weighted penalty function -- all configurable via a JSON specification with no code changes required. Key contributions include: a shift-window variable decomposition enabling mandatory break scheduling with centrality control; acuity-weighted workload equity; multi-granularity temporal resolution from 30 minutes to 2 hours; inter-week schedule stability; a grid-offset preprocessing technique for cross-midnight shifts; and a reproducible 36-configuration benchmark suite for community comparison. Evaluated on INRC-II benchmarks at both hourly and shift-level granularity and on 36 synthetic configurations.

cs.AI↗

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↗