Linear Programming

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4 papers in the last four weeks, with none the four weeks before. 0.0% of all new papers.

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Latest papers 29

Oct 1, 2026cs.LG

Linear Programming Representations and Strongly Polynomial Algorithms for Robust Markov Decision Processes

We study linear programming (LP) representations and strongly polynomial algorithms for robust Markov decision processes (RMDPs) with rational polyhedral state-action rectangular uncertainty in rewards and transitions. By encoding a finite sequence of robust policy-iteration steps, we construct a single LP whose optimal solutions recover the robust optimal value and all optimal stationary randomized policies. At fixed discount, the LP has polynomial dimension and encoding length and can be constructed in strongly polynomial time. We also develop a general complexity analysis of robust policy iteration that combines the cost of minimizing over uncertainty sets with the number of iterations needed to evaluate a policy. For a fixed discount factor, we use this analysis to improve the known complexity bounds for

\ell_1

and

\ell_\infty

RMDPs and establish new strongly polynomial bounds for general interval, weighted

\ell_1

, and Wasserstein RMDPs, as well as turn-based stochastic games with these uncertainty sets.

Han Zhong, Yinyu Ye

Shanghai Jiao Tong University · Shanghai Jiao Tong University, SIMIS, and Stanford University

Markov Decision Processes Linear Programming Decisions

Oct 1, 2026cs.LG

The Curvature of Regret in Contextual Linear Optimization

Decision-focused learning for linear optimization is complicated by the discontinuity of the optimizer, where small cost errors may leave the decision unchanged or move it to a different vertex. We show that this non-smooth pointwise behavior becomes locally quadratic after averaging over the data distribution, and we derive the curvature in closed form, specifically, a matrix-valued measure supported on the walls of the normal fan. This measure depends only on the feasible set, with the data distribution entering only as a weight. We then offer a tractable approximation for this curvature, computable with just one projection to the feasible set. We prove that the approximation weakly converges to the true population curvature. We offer one application of our findings, a decision-aware scenario generation method for expected-cost linear optimization. Our experiments test the quadratic and weak convergence laws and show a 30.8% regret improvement over uniform allocation on battery arbitrage.

Konstantinos Ziliaskopoulos, Alexander Vinel, Alice E. Smith

Auburn University Auburn, AL, USA · University of Alabama Tuscaloosa, AL, USA

Linear Programming Regret Scenario Generation

Oct 1, 2026math.OC

Reinforcement Learning to Accelerate Primal-Dual Hybrid Gradient for Linear Programming

Primal-dual hybrid gradient (PDHG) methods solve large-scale linear programs (LPs) using GPU-friendly matrix-vector products and projections, but their practical performance depends on coordinating algorithm parameters, acceleration, and restarts. We introduce GALLOP, which uses reinforcement learning to jointly learn continuous algorithm parameters and discrete restart decisions without differentiating through the solver. Its generalized accelerated PDHG update combines separate primal and dual extrapolation, history corrections, and restart anchoring with independently adjustable coefficients. We train a dimension-agnostic feedback policy using a groupwise proximal policy optimization objective that clips likelihood ratios separately for different control groups and excludes inactive acceleration controls on restart transitions. We evaluate GALLOP on six LP families and a public item-placement benchmark. On the main evaluation settings across the six families, GALLOP reduces iteration counts by factors of

1.9

-

5.6

and achieves up to a

16.0\times

speedup in algorithm wall-clock time over MPAX. With one policy trained per family, the learned policies generalize without retraining to within-family LPs

3\times

-

400\times

larger than the largest training instances, including Transport LPs with

10.24

million variables.

Jinhwan Sul, Alex Oshin, Evangelos A. Theodorou

Georgia Institute of Technology, USA

Linear Programming Primal-Dual Methods Proximal Policy Optimization Gradient

Sep 30, 2026cs.LG

Reformulation-Contrastive Learning for Mixed Integer Programs

Mixed-integer linear programs (MILP) model many real-world decision problems, motivating machine-learning methods that exploit recurring structure to accelerate MILP solving. MILPs can admit many equivalent formulations: integrality-preserving changes of variables and the addition of redundant constraints can alter their formulations while preserving the optimization problem. We leverage these reformulations as a source of self-supervision for learning general-purpose representations of MILP variables and constraints. We characterize the affine reformulations that are valid for every input instance, and distinguish re-descriptions, which leave variables unchanged, from substitutions, which transform them predictably. Building on equivariant self-supervised learning, we introduce ReMILP (reformulation-contrastive MILP representation learning), which jointly trains a graph neural network and a hypernetwork to predict how variable embeddings transform under changes of variables. Without solver-derived labels, ReMILP learns representations that exhibit the intended invariance and equivariance on unseen problem classes. Across binary solution, constraint activity and integrality gap prediction, these representations carry task-relevant information when frozen and provide a useful initialization for fine-tuning.

Ousema Bouaneni, Mathis Le Bail, Clément Elliker +2

LIX (École Polytechnique, IP Paris, CNRS) · AMIAD (Agence Ministérielle pour l’IA de Défense)

Mixed-Integer Programming Linear Programming Contrastive Learning Self-Supervised Learning

Sep 28, 2026cs.LG

Predictive Dual Smoothing for Column Generation

Solving large-scale linear programs efficiently is an important challenge in many optimization settings. A key technique is column generation, which alternates between solving the master problem over a restricted subset of the variables, and using a pricing subproblem to identify new variables to add. The pricing subproblem is guided by the dual solution of the current restricted master problem, but oscillations in these dual solutions can substantially slow convergence. Dual stabilization methods address this issue. Dual smoothing is a common stabilization method, which guides the pricing subproblem using a combination of the current dual solution and duals from previous iterations. However, while past dual solutions can stabilize the dual trajectory, they do not necessarily guide pricing towards useful new variables. We therefore introduce predictive dual smoothing, which instead combines the current dual solution with a learned prediction of future duals to steer pricing towards variables that are more useful in subsequent iterations. The predictor is trained offline using supervision extracted from standard column generation trajectories and is used only to modify the pricing subproblem's objective function, while exact reduced-cost checks and fallback pricing with the unsmoothed duals preserve correctness. Experiments on cutting stock and generalized assignment problems show that predictive dual smoothing substantially reduces generated columns and wall-clock time relative to standard column generation and existing classical and learned stabilization methods. These gains extend to out-of-distribution instance sizes, and predictive smoothing provides further improvements when combined with strong classical stabilization.

Senne Berden, Noah Schutte, Andrea Lodi +1

KU Leuven · TU Delft · Cornell Tech

Linear Programming Smoothing Stabilization Student-Generated Trajectories Data Generation

Sep 24, 2026cs.DS

A Nearly Quadratic Lower Bound for Linear Optimization over Convex Bodies in the Membership Oracle Model

We prove nearly quadratic lower bounds for randomized algorithms for linear optimization and uniform sampling over convex bodies in the membership oracle model. For linear optimization, this matches the known nearly quadratic upper bound up to a polylog factor in the dimension. For uniform sampling, this improves on the previous linear lower bound. Our construction also implies the same lower bound for volume estimation.

Santosh S. Vempala

Georgia Tech

Lower Bounds Linear Programming Convex Sets Polynomials Membership Inference Volume

Sep 23, 2026cs.LG

Resource-Adaptive Stochastic Gradient Descent for Online Linear Programming without Re-solving

The growth of large language model (LLM) inference and search services increases the scale of online linear programming problems, motivating computationally efficient algorithms. We develop resource-adaptive stochastic gradient descent (RASGD) for stochastic online linear programming. The algorithm uses one request and current inventory to update resource prices, requiring O(m) operations for m resources and memory per arrival and no LP or sample-average optimization. The central idea is to express the current-resource pricing logic of re-solving through a first-order SGD update: each arrival refreshes the remaining-inventory allowance in the dual objective, while the stepsize decreases for early learning and increases later to match the speed of inventory adjustment. Under standard non-degeneracy conditions, our algorithm is feasible on every sample path and achieves O(\log T) expected regret against the realized fractional hindsight optimum, which matches the lower bound, even for policies that know the distribution and have unrestricted computation. The analysis converts curvature around the fixed reference price into inventory stability without tracking optimal prices at changing resource levels. Numerical experiments show that RASGD achieves regret competitive with per-arrival LP re-solving and improves upon the tested first-order baselines, while retaining the computational efficiency of first-order methods. These results establish RASGD as a computationally efficient approach to achieving high allocation quality in large-scale OLP.

Jiameng Lyu

Department of Management Science, School of Management, Fudan University, Shanghai 200433, China

Linear Programming Learning-Augmented Algorithms Stochastic Optimization

Sep 21, 2026cs.LG

An Exact Junction-Tree Extended Formulation for Optimal Classification Trees

We develop an exact linear programming (LP) formulation for bounded-depth classification trees with binary features, using a junction-tree representation. The formulation is integral and supports recursive subtree optimization. Exact reductions make the model smaller while preserving the optimal value and recovery of an optimal tree. The reduced model supports two solution methods: column generation and message passing. Column generation solves integral restricted LPs and uses bounds over the full feasible domain to certify optimality. Message passing recursively combines optimal subtree costs. Both methods solve common subtree problems that, once the preceding tree decisions are fixed, can be evaluated independently and in parallel. Computational experiments show that the exact reductions substantially reduce the size of the junction-tree formulation. The resulting linear programming formulation certifies instances for which the tested mixed-integer formulation does not establish optimality within the same computational budget, while the column-generation and message-passing methods certify more instances and achieve an order-of-magnitude reduction in geometric-mean runtime relative to an existing state-of-the-art exact method for optimal classification trees.

Jiancheng TU, WenqiFan

Department of Computing, The Hong Kong Polytechnic University · Department of Management and Marketing, The Hong Kong Polytechnic University

Decision Trees Linear Programming Solvers

Sep 9, 2026cs.LG

Online Inverse Integer Linear Optimization via Small-Gradient Skipping: Constant Regret and Finite Mistakes

In online inverse linear optimization, the learner predicts a weight at each round, observes the optimal action of the agent, and updates its prediction. In the general setting, the gap of

\log T

between the regret upper bound

O(d \log T)

and the lower bound

Ω(d)

is unresolved (here

T

is the total number of rounds and

d

is the dimension). When the action set is M-convex, the regret is known to be bounded by

O(d \log d)

, but the method attaining it computes a center of gravity at every round. This paper therefore proposes Small-Gradient Skipping (SGS), a mechanism that skips the update at rounds without a mistake in the case where the correct action is uniformly separated from the other candidates, and applies it to online gradient descent, the online Newton step, and MetaGrad. The number of mistakes is then bounded, for all three, by a quantity independent of

T

; and for the online Newton step and for MetaGrad with SGS, the dimension dependence of the regret becomes

O(d^2)

when the forward problem is an integer linear program, that is, the factor

\log T

is removed. Moreover, when the action set is M-convex, the regret is bounded efficiently without computing a center of gravity.

Akira Kitaoka

Linear Programming Nonconvex Newton Skip

Jul 28, 2026cs.AI

Finding Optimal Cost-Bounded Plan Reductions: Refined Model

In some real applications a plan may later become unfeasible due to newly imposed budget constraints, yet, at the same time, using only the original actions of the plan and their order is mandatory. In this paper, we study the problem of extracting, from a precomputed plan, a valid subplan that maximizes utility while respecting a cost bound. Each goal is given a utility value and the plan is reduced by removing actions that support low-utility goals, while preserving both executability and the original action order. We show the decision variant is NP-complete and propose two exact methods to solve it: one via oversubscription planning (OSP) and another via Integer Linear Programming (ILP). This paper extends our previous work published at ICAPS 2026 (Del Toro, Fuentetaja, and García-Olaya 2026b). While the core framework remains as introduced there, we further introduce a refined ILP formulation that significantly decreases the model size and improves computational efficiency.

Martha Del Toro, Raquel Fuentetaja, Angel García-Olaya

Computer Science and Engineering Department, Universidad Carlos III de Madrid, Spain

Linear Programming Token Budget Allocation Optimality Subgoals Decisions

Jul 24, 2026math.OC

Explicit Iteration Complexity of Exact Data-Driven Inverse Optimization for Integer Linear Programs

A data-driven inverse optimization problem (DDIOP) is the problem of estimating the objective-function parameters (weights) that explain observed optimal-solution data, and it arises in many applications, including integer linear programming (ILP). It is known that, by applying gradient-based optimization methods to the suboptimality loss, the inverse optimization of ILPs can be solved exactly within finitely many oracle iterations, and that the required number of iterations is bounded as

T=O(1/γ(\ell_{\mathrm{sub}})^2)

in terms of a problem-dependent geometric constant

γ(\ell_{\mathrm{sub}})

. However, no means of bounding

γ(\ell_{\mathrm{sub}})

from below as a function of the problem size has been available, and hence the number of iterations could not be given as an explicit function of the problem size. We therefore give, when the forward problem is an integer linear program (ILP), the number of iterations sufficient for projected subgradient descent applied to the suboptimality loss to achieve exact consistency with the observed data, as a fully explicit function of the number of samples, the dimension of the features, the ranges of the features, and the structure of the constraint coefficient matrix, up to polynomial factors in the basic constants (the diameter of the weight set, the step-size parameter, and the Lipschitz constant of the suboptimality loss).

Akira Kitaoka

Linear Programming Optimal Sample Complexity Objective Functions

Jul 13, 2026cs.AI

LP Mining with LP2Graph: A Use Case for Railway Rescheduling

Like many optimization-driven domains, railway rescheduling relies on Mixed-Integer Linear Programming (MILP), yet the field's modeling knowledge is scattered across hundreds of papers in incompatible notations, and narrative surveys organize it subjectively: they classify models by vocabulary rather than by structure, and reproduce neither. We present LP Mining with LP2Graph, a method that mines the structure of published LP and MILP formulations into a reproducible dataset and an induced taxonomy. Its core, LP2Graph, represents each formulation admitted by its canonical grammar as a typed variable--equation graph derived from a single canonical model; once a source is extracted into that model, everything downstream is deterministic. Each source is parsed into this model, homologized, and clustered bottom-up (over variables, then constraints and the objective, then whole-model structure) and, separately, by application domain and solution approach; the resulting groups are labeled by a rule-seeded, self-updating classifier. We validate the representation rather than assume it: per-cluster representatives are regenerated as independent LaTeX and re-solved across CBC, HiGHS and Gurobi against the optimum reported in the source paper. The outcome is an objective, repeatable taxonomy of variables, constraints and model types: the principled foundation on which our raiLPminer line of automated railway-rescheduling model development builds.

Jörn Maurischat, Nikola Bešinović, Michael Färber

aTU Dresden, Chair of Railway Operations,"Friedrich List" Faculty of Transport and Traffic Sciences, Dresden, 01069, Germany · bTU Dresden, ScaDS.AI Dresden/Leipzig, Chair of Scalable Software Architectures for Data Analytics, Dresden, 01069, Germany

Mixed-Integer Programming Linear Programming Graph Representations

Jul 7, 2026cs.LG

GraphBU: MILP Instance Generation with Graph-Native Block Units

Mixed-integer linear programming (MILP) instances used for solver development are hard to obtain when models come from private or application-specific pipelines. A generator must keep the structure that solvers and learned policies rely on. Existing general generators usually choose their generation unit from a formulation template, summary statistics, local graph edits, or blocks found after recombination. These units do not explicitly record how a local part of the MILP is coupled to the rest of the instance. We propose GraphBU, a graph-native generator whose basic unit is a local subproblem plus its interface. The method promotes coupling nodes into master constraints or boundary variables and uses the resulting block units for compatibility-checked replacement. The analysis focuses on the properties needed by this construction: promotion separates interfaces, replacement can preserve feasibility under an interface-slack condition, and the graph construction is invariant to row-column permutations. On MILP instances generation, this unit keeps graph statistics close to the source family, preserves feasibility on most datasets, and improves downstream Predict-and-Search training. Genrated by GraphBU, The average graph-statistical similarity was approximately 0.934, the average feasibility was approximately 96.7%, and the average increase in the main index of downstream PS was approximately 8.0%.

Xiaolei Guo, Chenyu Zhou, Jianghao Lin +1

Shanghai Jiao Tong University

Mixed-Integer Programming Linear Programming Graph Representations

Jul 4, 2026cs.AI

Online Linear Programming for Multi-Objective Routing in LLM Serving

We study the online routing problem in large language model serving, where requests arrive sequentially and must be dispatched to parallel decode workers under tight batch-size and KV-cache constraints. Unlike widely used routing heuristics that are not tied to explicit service-level objectives (SLOs) and offer limited control over latency-throughput trade-offs, we introduce a multi-objective optimization framework that formulates routing as an online linear programming with interpretable decision rewards. We apply an efficient bid-price control policy based on the online linear programming that admits requests when their SLO-weighted benefit exceeds their shadow prices. To meet millisecond decision requirements, we develop a warm-started, projected first-order updates that track the evolving dual shadow prices online with predictable runtime. We integrate our router into the Vidur simulator and demonstrate substantial improvements over standard baselines across multiple SLO regimes, including end-to-end latency, time-to-first-token, throughput, and tail performance. A big picture from our result: a science-based approach outperforms others based on heuristics.

Zixi Chen, Yinyu Ye, Zijie Zhou

School of Mathematical Sciences, Peking University · Department of Industrial Engineering and Decision Analytics, HKUST

Large Language Model Routing Large Language Model Serving Linear Programming Vehicle Routing Problem

Jun 29, 2026cs.LG

Optimizing Nursing Care Taxi Dispatch Leveraging Integer Linear Programming Solvers and Machine Learning

In this paper, we formulate a new vehicle dispatch optimization problem, called Nursing Care Taxi Dispatch, as a variant of the Vehicle Routing Problem, considering constraints related to wheelchair use, user compatibility, pick-up and drop-off times, and vehicle limitations. Previous neural-based methods for Vehicle Routing Problems have typically addressed a few simple constraints, while our new problem involves multiple complex constraints, resulting in having fewer destinations to select. This complexity makes it more difficult to obtain solutions that allow all nodes to be visited with a limited number of vehicles. To balance low violation rate, computational efficiency, and solution quality, we propose a supervised machine learning approach based on the Transformer architecture. We first obtain a set of high-quality solutions using an integer linear programming solver for given inputs and then train our learning model through supervised learning. Additionally, we introduce the post-processing of the paths generated by the learning model, ensuring that all constraints are satisfied. We compared each instance's objective function value (operating time), execution time, and constraint violation rate across different methods: our proposed method and some existing methods including integer linear programming and machine learning-based methods, using real-world facility data. Our method successfully produced balanced solutions regarding operating time, execution time, and constraint violation rate. Notably, we observed a decrease in the operating time for all problem sizes and regions, while keeping constraint violations to a minimum compared to existing methods. Especially, the decrease reached up to 8% for problem sizes with fewer than 30 users.

Riku Nakao, Akihito Hiromori, Hamada Rizk +1

The University of Osaka, Japan · RIKEN R-CCS, Japan · Tanta University, Egypt

Vehicle Routing Problem Mixed-Integer Programming Linear Programming Emergency Departments Structural Constraints

Jun 18, 2026math.OC

Optimal Coarse Correlated Equilibria in Mean Field Games: Linear Programming and No-Regret Learning

We introduce optimal coarse correlated equilibria for continuous-time mean field games. A coarse correlated equilibrium is a randomized recommendation scheme from which no player can gain by ignoring the recommendation and switching to an alternative strategy. The problem is as follows: a moderator selects, among all mean-field coarse correlated equilibria, one that optimizes a prescribed performance criterion, which may differ from the representative player's objective. After formulating the problem, we develop a linear programming (LP) formulation, prove the existence of optimal LP coarse correlated equilibria, and relate the LP characterization to the original probabilistic setting. Building on this characterization, we design a no-regret primal-dual algorithm, based on an equivalent Lagrangian formulation of the external-regret constraint, for learning such equilibria. We provide explicit convergence rates for the learning algorithm, and numerical examples illustrate the method.

Luciano Campi, Federico Cannerozzi, Ioannis Tzouanas

DEPARTMENT OF MATHEMATICS “FEDERIGO ENRIQUES”, UNIVERSITY OF MILAN, VIA SALDINI 50, 20133, MILAN, ITALY. · CENTER FOR MATHEMATICAL ECONOMICS (IMW), BIELEFELD UNIVERSITY, BIELEFELD, GERMANY.

Mean Field Games Equilibrium Mean-Field Limit Plan Regret Linear Programming Recommendation Quality

Jun 15, 2026cs.LG

LiFT: Local Search via Linear Programming for Overfitting-Controlled Transformers

This paper proposes a Linear Programming (LP)-based local search framework for fine-tuning pretrained transformer models with explicit control against overfitting. The approach formulates transformer fine-tuning as a bilevel optimization-based regularization problem, in which model parameters and regularization hyperparameters are jointly updated. Information collected during initial warm-up iterations, including validation gradients and training Hessian information, is used to construct a local descent direction by solving an LP that minimizes a scaled directional derivative while preserving training optimality. This validation-aware descent direction enables focused local updates of both parameters and regularization hyperparameters, reducing overfitting without requiring repeated full retraining cycles. The resulting method, termed Linear Programming-based Fine-Tuning (LiFT) for transformers, differs from conventional fine-tuning by systematically identifying task-specific updates rather than relying on heuristic or grid-based hyperparameter selection. Experiments on GPT-2 Small fine-tuned on WikiText-2 demonstrate that LiFT enables effective adaptation through selective tuning of transformer blocks and regularization parameters, yielding consistent improvements in test perplexity across multiple layer configurations and regularization settings, with particularly pronounced gains in overfitting-prone scenarios. Beyond empirical performance, LiFT establishes a principled connection between transformer fine-tuning, bilevel optimization, local search, and regularization theory.

Abhishek Shukla, Anikeit Khanna, Ankur Sinha +1

Department of Management Sciences, Indian Institute of Technology Kanpur, Kanpur-208016, Uttar Pradesh, India · Brij Disa Centre for Data Science and AI, Indian Institute of Management Ahmedabad, Ahmedabad-380015, Gujarat, India · Department of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur-208016, Uttar Pradesh, India +1

Parameter-Efficient Fine-Tuning Methods Model Fine-Tuning Transformer Architectures Regularization Large Neighborhood Search Linear Programming

Jun 7, 2026cs.LG

Scaling Decision-Focused Learning to Large Problems with Lagrangian Decomposition

Decision-focused learning has shown great promise for addressing predict-then-optimize problems, particularly in the presence of under-specified models. However, its practical deployment is often hindered by high computational costs and limited scalability, as it requires solving a constrained optimization problem for each training instance at every iteration. To address these challenges, we propose a novel framework that incorporates Lagrangian decomposition into the decision-focused learning paradigm. Specifically, we introduce a new surrogate objective along with two loss functions for evaluating and training the underlying prediction model. We further propose two variants of our approach, which offer different trade-offs between computational efficiency and solution quality. Our framework can be seamlessly integrated with standard decision-focused learning methods, including Smart Predict-then-Optimize (SPO+) and Implicit Maximum Likelihood Estimation (IMLE). Through experiments on two standard benchmarks, the multi-dimensional knapsack problem and quadratic portfolio optimization, we demonstrate that our approach achieves competitive performance while remaining amenable to parallelization. In particular, it consistently outperforms traditional decision-focused learning methods on large-scale instances, involving up to eight times more variables than those typically considered in related work. The implementation is available at https://github.com/corail-research/DFL-LD.

Stéphane Eilles-Chan Way, Hugo Percot, Quentin Cappart +2

Polytechnique Montréal, Montreal, Canada · Ecole Polytechnique, Palaiseau, France · UCLouvain, Louvain-la-Neuve, Belgium +2

Linear Programming Decentralized Optimization Portfolio Optimization Decomposition Lagrangian Methods Scaling

Jun 7, 2026math.OC

Parameter Tuning with Generalization Guarantees for GPU-Accelerated Linear Programming

Recent research has developed practical, parallelizable first-order methods for large scale linear programming, but performance is highly dependent on hyperparameter selection. We derive generalization guarantees for hyperparameter tuning within (cu)PDLP, a state-of-the-art first-order LP solver designed for modern hardware. First, we pin down the behavior of PDHG, the primal-dual hybrid gradient algorithm that underlies PDLP, as a function of its step size and primal weight, leading to linear sample complexity guarantees for learning those parameters. We then conduct a structural analysis of PDLP, which augments PDHG with several specialized techniques like preconditioning, adaptive step sizes, averaging, adaptive restarts, and smoothed primal weight updates. Our analysis captures the behavior of the solution trajectory as a function of the hyperparameters and leverages recent advances in data-driven algorithm design to obtain polynomial sample complexity guarantees for learning those hyperparameters. Finally, we conduct proof-of-concept experiments that demonstrate the need for data-driven PDLP parameter tuning. Our results showcase the versatility of the data-driven algorithm design toolkit for principled hyperparameter tuning within solver-grade implementations of complex modern optimization algorithms.

Siddharth Prasad, Dravyansh Sharma

Toyota Technological Institute at Chicago

Linear Programming Primal-Dual Methods Tuning Optimal Sample Complexity

May 29, 2026cs.NE

Linear Ordering Problem: Time for a Change

The Linear Ordering Problem (LOP) is a fundamental combinatorial optimization problem with important applications in areas such as economics, social choice, and machine learning. Its most prominent use is the triangulation of economic input-output tables, which helps identify critical industries in an economy. Most existing algorithms have been evaluated on benchmarks derived from outdated macroeconomic data, which no longer reflect the structure of contemporary economies. Furthermore, LOP instances often exhibit many distinct global optima that can differ substantially from one another, creating challenges for applications that rely on a single solution. To address these limitations, we introduce a novel benchmark suite derived from up-to-date real-world economic data and an algorithmic scheme that leverages state-of-the-art LOP metaheuristics to generate diverse sets of high-quality solutions, together with metrics for assessing both quality and diversity. Experiments were conducted to report results on the proposed benchmark suite under both the traditional single-solution setting and the newly introduced multi-solution scenario

Fabrizio Fagiolo, Marco Baioletti, Valentino Santucci

University for Foreigners of Perugia, Perugia 06123, Italy · University of Perugia, Perugia 06123, Italy

Linear Programming Metaheuristics Economies Change

May 28, 2026cs.LG

Solving Integer Linear Programming with Parallel Tempering

Integer Linear Programming (ILP) serves as a versatile framework for modeling a wide range of combinatorial optimization problems, typically addressed by sophisticated exact solvers or heuristics. While learning-based approaches have recently shown their effectiveness, they suffer from poor generalization to out-of-distribution instances and inherent dependence on external solvers. In this work, we propose a solver-free, sampling-based optimization framework for ILP that directly explores discrete feasible regions without training or external solvers. Exploiting the linear structure of ILP, we employ a Locally-Balanced Proposal to construct a transition kernel, thereby avoiding the gradient approximation. To overcome the highly multimodal nature of ILP energy landscapes, we integrate Parallel Tempering. In addition to standard temperature tempering, we introduce penalty tempering, which modulates constraint barriers while preserving the objective landscape over feasible solutions. Empirically, our method consistently outperforms SCIP across all four benchmarks, matches or exceeds Gurobi on two of four tasks within a 200-second budget, and is substantially more robust to distribution shift than learning-based methods. Furthermore, on MIPLIB 2017 instances, our framework remains competitive with classical solvers without any problem-specific tuning.

Kyuil Sim, Sanghyeok Choi, Jinkyoo Park

1KAIST · University of Edinburgh

Mixed-Integer Programming Linear Programming Free-Energy Landscape

May 26, 2026cs.AI

Developing a Totally Unimodular Linear Program for Optimal Conformance Checking: When and Why It Complements A*

Alignment-based conformance checking is the state-of-the-art approach for comparing observed process executions with normative process models. The standard exact solution relies on an A*-based heuristic search, which can exhibit exponential runtime in the presence of long traces or substantial deviations. This paper introduces a reformulation of alignment-based conformance checking as a totally unimodular linear program (LP) defined on the reachability graph of the synchronous product. By exploiting the underlying network-flow structure, the proposed formulation guarantees the existence of an integral optimal extreme-point solution through LP relaxation, thereby avoiding the combinatorial overhead associated with integer variables and branch-and-bound search. We conduct an extensive empirical evaluation on more than 2.1 million conformance checking instances derived from real-world and synthetic benchmark datasets. The results show that A* and the LP approach exhibit complementary performance characteristics: the former performs best on short, well-conforming traces, while the LP formulation provides substantial speedups for longer traces with deviations, precisely where conformance checking is most informative. Based on these findings, we derive simple algorithm-selection guidelines that combine both approaches, achieving average runtime savings of 38.6% with 96% selection accuracy compared to always using A*.

Izack Cohen

aFaculty of Engineering, Bar-Ilan University, Ramat Gan, Israel

Probabilistic Model Checking Linear Programming Reachability Traces

May 14, 2026stat.ML

From Data to Action: Accelerating Refinery Optimization with AI

Nowadays refinery optimization utilizes sheer amounts of data, which can be handled with modern Linear Programming (LP) software, but the interpreting and applying the results remains challenging. Large petrochemical companies use massive models, with hundreds of thousands of input matrix elements. The LP solution is mathematically correct, but simplifications are made in the model, and data supply errors may occur. Therefore, further insight is needed to trust the results. The LP solver does not have a memory, so additional understanding could be gained by analyzing historical data and comparing it to the current plan. As such, machine learning approaches were suggested to support decision making based on the LP solution. Among these, Anomaly Detection tools are proposed to be used in tandem with the LP output. A transformed version of the popular ECOD methodology is applied. New methods are proposed to handle high-dimensional data: choosing the most informative pairs. Then, this is used alongside two 2D Anomaly Detection algorithms, revealing several business opportunities and data supply errors in the MOL refinery scheduling and planning architecture.

Dániel Pfeifer, Ábrahám Papp, Tibor Bernáth +4

aBudapest University of Technology and Economics, Műegyetem rkp. 3., Budapest, 1111, Hungary · bMOL Group, Dombóvári út 28., Budapest, 1117, Hungary

Linear Programming Optimization Modeling Ecosystems

Apr 27, 2026cs.GT

Asymmetric-Information Resource Allocation Games: An LP Approach to Purposeful Deception

In this work, we introduce the Deceptive Resource Allocation Game (DRAG), which studies purposeful deception within a Bayesian game framework. In DRAG, a Defender allocates resources across the true asset and several decoys to influence an Attacker's beliefs and actions, with the goal of diverting the Attacker away from the true asset. We seek to characterize purposeful deception, whereby the Defender deceives only when doing so improves its performance. To this end, we solve for the Perfect Bayesian Nash Equilibrium (PBNE) of the corresponding game. We show that, despite the coupled belief-policy interdependence, the problem admits an efficient, non-iterative linear programming formulation. Numerical results demonstrate that the resulting policies naturally balance effective allocation and belief manipulation, giving rise to purposeful and emergent deceptive behaviors.

Longxu Pan, Yue Guan, Daigo Shishika +1

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, USA · College of Engineering and Computing, George Mason University, Fairfax, VA, USA

Deception Linear Programming

Apr 5, 2026cs.LG

Learning an Interpretable Risk Scoring System for Maximizing Decision Net Benefit

Risk scoring systems are widely used in high-stakes domains to assist decision-making. However, existing approaches often focus on optimizing predictive accuracy or likelihood-based criteria, which may not align with the main goal of maximizing utility. In this paper, we propose a novel risk scoring system that directly optimizes net benefit over a range of decision thresholds. The model is formulated as a sparse integer linear programming problem which enables the construction of a transparent scoring system with integer coefficients, and hence, facilitates interpretation and practical application. We also establish fundamental relationships among net benefit, discrimination, and calibration. Our analysis proves that optimizing net benefit also guarantees conventional performance measures. We evaluated our method on multiple public datasets as well as on a large-scale credit risk dataset. This computational study demonstrated that our interpretable method can effectively achieve high net benefit while maintaining competitive discrimination and calibration performance.

Wenhao Chi, Ş. İlker Birbil

Yau Mathematical Sciences Center, Tsinghua University · Amsterdam Business School, University of Amsterdam

Risk Stratification Utility Maximization Decision Thresholds Linear Programming Decisions

Jun 2, 2025cs.LG

Efficient Learning of Balanced Signed Graphs via Sparse Linear Programming

Signed graphs are equipped with both positive and negative edge weights, encoding pairwise correlations as well as anti-correlations in data. A balanced signed graph is a signed graph with no cycles containing an odd number of negative edges. Laplacian of a balanced signed graph has eigenvectors that map via a simple linear transform to ones in a corresponding positive graph Laplacian, thus enabling reuse of spectral filtering tools designed for positive graphs. We propose an efficient computation method to learn a balanced signed graph Laplacian directly from data. Specifically, extending a previous linear programming (LP) based sparse inverse covariance estimation method called CLIME, we formulate a new LP problem for each Laplacian column

i

, where the linear constraints restrict weight signs of edges stemming from node

i

, so that nodes of same / different polarities are connected by positive / negative edges. We derive a feasible CLIME parameter

ρ_i

for each sign-constrained column problem. We solve the LP problem efficiently by tailoring a sparse LP method based on ADMM. We theoretically prove that the row / column updates produce a non-increasing objective sequence, and show that the iterations are terminated in a finite number of steps. Extensive experimental results on synthetic and real-world datasets show that our balanced graph learning method outperforms competing methods and enables reuse of spectral filters, wavelets, and graph neural nets (GNN) constructed for positive graphs.

Haruki Yokota, Hiroshi Higashi, Yuichi Tanaka +1

Graph Laplacians Spectral Filtering Balanced Learning Linear Programming Covariance

Aug 1, 2024cs.DS

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.

Guokai Li, Zizhuo Wang, Jingwei Zhang

School of Data Science, The Chinese University of Hong Kong, Shenzhen, Guangdong, 518172, P.R. China

Linear Programming \Widetilde{\Mathcal{O}}(\Sqrt{T})$ Regret Free

Mar 28, 2024math.OC

Fisher-Rao Gradient Flows of Linear Programs and State-Action Natural Policy Gradients

Kakade's natural policy gradient method has been studied extensively in recent years, showing linear convergence with and without regularization. We study another natural gradient method based on the Fisher information matrix of the state-action distributions which has received little attention from the theoretical side. Here, the state-action distributions follow the Fisher-Rao gradient flow inside the state-action polytope with respect to a linear potential. Therefore, we study Fisher-Rao gradient flows of linear programs more generally and show linear convergence with a rate that depends on the geometry of the linear program. Equivalently, this yields an estimate on the error induced by entropic regularization of the linear program which improves existing results. We extend these results and show sublinear convergence for perturbed Fisher-Rao gradient flows and natural gradient flows up to an approximation error. In particular, these general results cover the case of state-action natural policy gradients.

Johannes Müller, Semih Çaycı, Guido Montúfar

Department of Mathematics, RWTH Aachen University, Aachen, 52062, Germany · Departments of Mathematics and Statistics & Data Science, University of California, Los Angeles, 90095, USA · Max Planck Institute for Mathematics in the Sciences, Leipzig, 04103, Germany

Policy Gradient Fisher Information Matrix Wasserstein Gradient Flows Gradient Linear Programming Polytopes

Aug 11, 2021cs.AI

Stable Marriage Problems with Ties and Incomplete Preferences: An Empirical Comparison of ASP, SAT, ILP, CP, and Local Search Methods

We study a variation of the Stable Marriage problem, where every man and every woman express their preferences as preference lists which may be incomplete and contain ties. This problem is called the Stable Marriage problem with Ties and Incomplete preferences (SMTI). We consider three optimization variants of SMTI, Max Cardinality, Sex-Equal and Egalitarian, and empirically compare the following methods to solve them: Answer Set Programming, Constraint Programming, Integer Linear Programming. For Max Cardinality, we compare these methods with Local Search methods as well. We also empirically compare Answer Set Programming with Propositional Satisfiability, for SMTI instances.

Selin Eyupoglu, Muge Fidan, Yavuz Gulesen +5

Computer Science, Faculty of Engineering and Natural Sciences, Sabanci University, Istanbul, Turkey · Economics, Faculty of Arts and Social Sciences, Sabanci University, Istanbul, Turkey

Satisfiability Linear Programming Answer Set Programming Cardinality Preference Alignment Learning Empirical Study