Mixed-Integer Programming

Operational Air Traffic Flow and Capacity Management (ATFCM) balances flight demand with available sector capacity, to ensure safe and efficient operations. Mathematical models enhance operational ATFCM performance by framing demand-capacity balancing as an optimization problem, maximizing efficiency while adhering to safety constraints. However, SOTA research optimizes the aircraft trajectories (called ATFM) or the sector configuration (called DAC) separately. This leaves a research gap of whether joint optimization of ATFM and DAC can bring benefits. We partially address this limitation by introducing a joint ATFCM model with an encoding in Answer Set Programming (ASP). The ASP implementation is evaluated against two baselines applied to our joint model: a SOTA Mixed Integer Programming (MIP) model and an iterative CASA-based heuristic. Computational experiments utilize an instance generator fitted to historical OpenSky Network flight data. Our results indicate that the ASP model outperforms the MIP model, while ASP remains competitive against heuristics on small instances. Furthermore, while DAC has the largest improvement on solving performance compared to rerouting and delaying, unrestricted variants of DAC or rerouting lead to search space thrashing.

Spacecraft operations scheduling is a highly constrained, long-horizon combinatorial optimization problem that traditionally relies on heuristics, constraint programming, or manual planning. We present a scalable deep reinforcement learning framework developed and deployed for NASA's Carruthers Geocorona Observatory mission. Our framework introduces a macro-action abstraction known as activity blocks coupled with dynamic action-masking to navigate the intractably large search space and strictly enforce complex power, thermal, and instrument constraints. The resulting architecture generates globally feasible schedules with overwhelming probability, establishes operational trust, and executes a full training cycle in under six hours, circumventing the need for policy robustness by enabling rapid, on-demand retraining. Further, resulting schedules outperform baseline heuristics in scheduled science quality. The deep reinforcement learning framework was deployed as the default operational scheduler for the Carruthers Geocorona Observatory mission from the outset of the mission, demonstrating that deep reinforcement learning can be trusted for real spacecraft operations under complex, evolving constraints.

Neural combinatorial optimization (NCO) has shown that policies trained by reinforcement can construct strong solutions to NP-hard problems directly from raw instances. What such a policy actually learns, as opposed to what its decoder expresses, remains much less clear. We study this distinction on the vertex-guard Art Gallery Problem, the NP-hard task of choosing polygon vertices from which to observe an entire region. A pointer-network policy is trained from a coverage-aware reward over its own rollouts under the constraint we call geo-free inference: at test time it sees only vertex coordinates, with no visibility computation and no geometric oracle. The policy places guards economically but leaves a tail of under-covered polygons that widens far beyond the training range. To locate the cause, we freeze the trained encoder and read its embeddings with a small single-shot classifier, still geo-free at inference. The classifier closes most of the feasibility gap, in and out of distribution and at up to roughly five times the training range, cutting under-covered polygons by about an order of magnitude at an explicitly reported cost in guard count. We read this as evidence that the reinforcement-trained representation already encodes the geometry required for feasibility, and that residual failures reflect decoder calibration rather than missing knowledge. Probing a frozen encoder thus offers a practical way to ask what a neural combinatorial solver has internalized.

Large-scale optimization problems are often solved repeatedly under similar structural conditions, leading to substantial computational overhead. This occurs in applications such as power systems, transportation, and supply chain networks, where the underlying structure is fixed while parameters frequently vary under perturbations. This paper proposes a Learning to Optimize (LTO) framework that accelerates the solution of large-scale general mixed-integer problems by leveraging the concept of a backdoor, i.e., a subset of variables that drive most of the computational complexity. The proposed BIPC framework consists of three phases. Phase I is an identification procedure that discovers a backdoor for a set of instances in the distribution. Phase II uses supervised learning to develop machine learning models that, given an instance, predict values for bounded-domain backdoor variables and intervals for wide-domain backdoor variables. These predictions define a reduced optimization problem where the predictions constrain the backdoor variables, while the other variables remain free. Phase III optimizes this reduced problem and, if necessary, applies a correction step to restore feasibility or the optimality guarantees. Experiments on real-world, large-scale problems show substantial reductions in solution time with only a limited loss in solution quality. The framework enables organizations to solve large-scale optimization problems efficiently in the presence of frequent perturbations, such as unexpected events, demand fluctuations, or operational changes. Because these changes affect parameters rather than the problem structure, BIPC can quickly provide high-quality, feasible solutions, offering a practical approach to integrating machine learning into existing optimization pipelines.

Virtual training environments are software-intensive systems in which reinforcement learning (RL) agents learn, adapt, and demonstrate meaningful behavior. Virtual training environments offer a safe and cost-efficient alternative to training agents in real-world settings. However, to converge, most realistic RL problems require training in multiple, mostly similar but slightly different environments - i.e., families of environment variants. The typical development process of environment families is a labor-intensive and error-prone manual endeavor that does not scale well. To alleviate these issues, in this paper, we propose a model-driven approach for developing families of RL training environments. To obtain the family of environments, we develop an approach and prototype tool. In our approach, a hybrid genetic algorithm - a combination of population-based global search and heuristic local search - generates environment families. Mutations and constraints are expressed as model transformations and are operationalized into a search process by a state-of-the-art model transformation engine. We demonstrate the soundness of our approach in a wildfire mitigation scenario and curriculum learning - a particular learning paradigm that relies on environment families.

This paper introduces RACL, a Reasoning-Agent Control Layer for metaheuristics. RACL places a reasoning agent above an existing optimizer. The agent does not replace the optimizer and does not modify business constraints. Instead, it controls the optimizer's internal search behavior by observing operational memory, reasoning over past behavior, formulating bounded hypotheses, testing interventions, evaluating outcomes, applying guardrails, consolidating useful policies and explaining its decisions. The experiment uses vehicle routing as a testbed, but the contribution is not a new routing solver, a particular ALNS configuration or a specific set of routing rules. The contribution is the RACL method: a way for a reasoning agent to discover, validate, consolidate and explain algorithmic control rules for a metaheuristic. In the current experimental setting, RACL improves or ties the Operational Memory Policy in 21 of 21 feasible cases and improves or ties a non-reasoning Stagnation-Triggered Policy in 18 of 21 feasible cases, with an average RACL vs STP cost delta of -0.641%. In the Sevilla-9/10 runtime sample, RACL improves average cost by -8.337% versus Fixed and -1.605% versus STP without showing material computational overhead. During the proof-of-concept, Codex was used as an in-the-loop reasoning agent observing executions, interpreting logs and proposing live bounded interventions. The policy proxy was later used only to make quantitative evaluation reproducible.

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.

We develop a structure-oriented randomized neural network framework, termed SO-RaNN, for the Poisson-Nernst-Planck (PNP) system and the Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system. The decoupled linearized subproblems are solved iteratively by randomized neural networks in a space-time framework. For the concentration variables, a pointwise cut-off is used to enforce positivity at the value level, and discrete mass-scaling factors are computed at selected correction instants and interpolated in time, so as to ensure exact mass matching at those instants and to promote approximate mass preservation between them. To introduce an auxiliary discrete dissipation mechanism, we further employ an SAV-type post-processing correction, which yields monotonicity of the SAV auxiliary variable under the ideal SAV update. For the PNP-NS system, a structure-preserving randomized neural network (SP-RaNN) is used for the velocity field, so that the velocity approximation satisfies the incompressibility constraint pointwise by construction. On the theoretical side, we derive residual-based estimates for the raw, uncorrected RaNN solvers of the linearized subproblems, formulate a conditional local-in-time convergence result for the raw outer Picard iteration of the PNP system, and analyze the value-level positivity correction together with the mass-correction and SAV post-processing steps. For the PNP-NS system, we establish an approximation result for the SP-RaNN space and provide a conditional error statement for the corresponding linearized Oseen-type problem. Numerical experiments demonstrate approximation accuracy in the source-driven manufactured tests and illustrate the intended value-level positivity correction, selected-time mass matching, computed free-energy curves based on the final gauge-fixed potential, and divergence-free approximation in benchmark tests.

Formal tools such as SAT and SMT solvers are increasingly embedded in language model reasoning pipelines when a safety or security critical question can be formulated in logic. Unlike chain of thought whose steps are sampled from the model distribution without formal guarantee, a solver produces a sound and independently verifiable answer. However, the soundness guarantee can be lost in the interaction between the solver and the model. The hybrid pipeline has three components: formalizing the question, deciding it, and narrating the result. Prior work has studied the formalization and decision, but not narration, which is the step that turns a formal tool's output into the user answer. To fill the narration gap, we first model the LLM-solver loop as a verified decision procedure. We further evaluate five open-sourced models under prompt injection, and we find certificate gating makes the solver verdict sound, while an adversary can invert a verified conclusion across phrasings and channels. We study the mitigation through hardened prompt that reduces injection significantly but cannot eliminate it and still suffers under adaptive attack. Combining the formal analysis and empirical studies, we show in the LLM-solver loop, robustness does not reach to the answer that the user finally reads.

This work presents a tool for the synthesis and simulation of probabilistic architectures for solving combinatorial optimization problems by mapping them to the Ising model. The proposed approach automatically constructs the Ising Hamiltonian and determines the number of probabilistic elements (p-bits) based on problem characteristics such as size and topology. Furthermore, the tool introduces an adaptive strategy for selecting the most suitable update algorithm among Gibbs Sampling, Simulated Annealing (SA), Simulated Quantum Annealing (SQA), and cluster-based methods. Experimental results using benchmark problems demonstrate improved convergence behavior and flexibility compared to fixed approaches. The proposed framework enables systematic evaluation of probabilistic computing strategies and supports the development of future hardware implementations based on MTJs and p-bits.

The Traveling Salesman Problem (TSP) is a cornerstone of combinatorial optimization and arises in many practical scenarios. Although graph-based learning approaches have been explored for TSP, the question of how to exploit graph structure more effectively remains open. We present the Anisotropic Graph Diffusion Network (AGDN), a new Graph Neural Network framework designed to solve TSP. Our method tackles two central difficulties: (1) the lack of informative topological prior in fully connected TSP graphs, and (2) losing connected nodes in the optimal solution after the commonly used graph sparsification techniques. To overcome these issues, we construct a MixScore transition matrix that merges node similarity with pairwise distance, and we develop an anisotropic graph diffusion strategy that supports efficient information exchange across multiple hops. Comprehensive experiments spanning diverse instance sizes and node distributions show that AGDN consistently outperforms existing methods while keeping computation time competitive. Furthermore, AGDN generalizes well to problem sizes and distributions beyond those seen during training. The implementation is publicly available at: https://github.com/LabRAI/AGDN.

Neural combinatorial optimization (NCO) offers a promising alternative to traditional heuristic-based methods for solving complex graph optimization problems by proposing to learn heuristics through data. This class of problems frequently arises in automation, as it can be used to model a variety of applications. While NCO has been extensively studied for deterministic combinatorial optimization problems, there are only a few works that aim to solve stochastic combinatorial optimization problems. In this work, we present N(CO)$^2$: Neural Combinatorial Optimization with Chance cOnstraints to solve the Stochastic Orienteering Problem (SOP) without the use of hand-crafted heuristics. By integrating a reinforcement learning (RL) framework, the model optimizes path selection under uncertainty, effectively balancing exploration and exploitation. Empirical results demonstrate that our method generalizes well across diverse SOP instances, achieving competitive performance compared to the state-of-the-art mixed-integer linear program (MILP) for the task. The proposed approach reduces human effort in heuristic design while enabling adaptive and efficient decision-making in uncertain environments.

Evolutionary algorithms have been used for a wide range of multi-objective combinatorial optimization problems. Despite practical success, theoretical results on the runtime of evolutionary algorithms for multi-objective combinatorial problems are rather limited. One classical problem that has been investigated is the multi-objective minimum spanning tree problem for which runtime bounds have been obtained to compute all extremal corner points of the Pareto front. With this paper, we provide some more detailed insights into the structure of the Pareto front when the edge weights take on a small number of distinct values. Based on these insights, we derive new runtime results for evolutionary multi-objective algorithms and complement our theoretical results with experimental investigations.

In skill-constrained production-inventory systems, the qualified human capacity available tomorrow depends on training decisions made today: production requires certified workers, certifications decay unless maintained, and training consumes the same scarce worker hours that production needs now. We study a closed-loop skill-constrained model predictive controller that, at every shift, solves a finite-horizon mixed-integer program over production, inventory, backlog, and training, with binary predicted certification, hard production eligibility, and an interpretable terminal value that prices certified-capacity gaps at the horizon boundary; only the first-period action is applied before replanning. On synthetic, seed-controlled SkillChain-Gym scenarios - announced and surprise new-skill shocks, demand shocks, absenteeism, forecast- and availability-quality modes, capacity-boundary and training-rate sweeps, and negative controls - we evaluate the controller against production-only and maintenance-only ablations, static cross-training insurance plans, and a strong reactive heuristic, under an ex-ante locked configuration and paired statistics. The result is regime dependence, not superiority: no policy class dominates. Predictive control helps when skill or labor bottlenecks are forecastable early enough for training to complete; lean static insurance remains hard to beat under surprise shocks, near the demand-capacity boundary, and wherever pre-shock slack makes insurance cheap. Attribution ablations separate certification maintenance, re-acquisition of lapsed certifications, and greenfield skill acquisition. Forecastability, not adaptivity per se, decides when predictive control pays.

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.

Finding D-optimal designs for generalized linear models (GLMs) is challenging due to the dependence of the Fisher information matrix on unknown parameters and the lack of closed-form solutions, particularly when input factors include both discrete and continuous variables. Although classical algorithms and recent metaheuristic approaches have offered partial solutions, there remains a need for robust and computationally efficient methods. In this paper, we propose a penalized Particle Swarm Optimization (PSO) approach, named $p$-PSO. Here we introduce a new, general-purpose penalty formulation for constrained optimization and demonstrate its effectiveness in optimal design problems. The formulation is algorithm-agnostic and applicable to a broad class of black-box optimization methods. Results show that the method is highly efficient, with its primary contribution being a penalty formulation that enables the direct use of an off-the-shelf PSO algorithm and extends naturally to more general constrained optimization tasks.

CMA-ES is, per run, a local optimizer; multimodal search relies on restart strategies such as IPOP and BIPOP, which draw every restart uniformly and reuse no information from previous evaluations. Multi-Start Clustering CMA-ES (MSC-CMA-ES) makes restarts structure-aware: in alternating cycles, a Sobol pre-sample is partitioned into approximate basins of attraction by nearest-better clustering, restarts are seeded basin by basin with locally scaled step sizes and population sizes, redundant basin visits are detected and excluded, and the remaining budget is spent on an unbounded local refinement of the best-so-far solution. We evaluate the method on four CEC suites (CEC2014, CEC2017, CEC2020, CEC2022) at their official budgets, across ten (suite, dimension) cells with dimensions 5-30, 51 runs per function, against BIPOP-CMA-ES and five differential-evolution algorithms (ARRDE, jSO, j2020, NL-SHADE-RSP, LSRTDE). Read per function class, MSC-CMA-ES leads on one class, is mixed on a second, and trails on the third. On composition functions, MSC-CMA-ES attains the best value on all four aggregate measures, with 2.7x the fixed-budget target coverage of BIPOP-CMA-ES - the highest composition coverage of any algorithm evaluated. On basic functions, it achieves the best (lowest) median error but exhibits a lower deep-target coverage - the measured price of spending budget on landscape discovery. On hybrid functions both CMA variants trail the leading DE algorithms; the deficit belongs to the CMA family, not to the restart mechanism. All results and scripts are publicly available.

The permutation flow shop scheduling problem (PFSP) is a classical NP-hard combinatorial optimization problem in intelligent manufacturing. In practice, PFSP is commonly addressed using metaheuristic algorithms, among which the iterated greedy (IG) algorithm is widely adopted due to its simplicity and strong empirical performance. However, classical IG relies on a single fixed destruction operator, which often limits exploration and leads to search stagnation on large and complex problem instances. To address this issue, this work proposes a multi-operator IG algorithm, termed IG-DOE, which enhances exploration by switching among heterogeneous destruction operators along a single search trajectory. The core mechanism, called stagnation-triggered sequential switching, activates the next destruction operator in an ordered destruction operator ensemble (DOE) when stagnation is detected, thereby enriching the perturbation behavior of classical IG. Moreover, to reduce reliance on expert-crafted operators, a large language model (LLM)-assisted framework, termed SCOE, is introduced to automatically construct a high-quality DOE through stagewise evolution, state-awareness, and cooperative evaluation. Experiments on the challenging VRF-hard-large benchmark show that the DOE evolved from smaller problem instances generalizes well to larger unseen instances. Under the same CPU-time limit, IG-DOE obtained much better average performance than QIG, a state-of-the-art IG algorithm. Additional experiments on real-world industrial-data-derived instances further show that the evolved DOE can generalize effectively to different data distributions without additional adaptation.

Population-based and distributional optimization methods, from evolution strategies and consensus-based optimization to covariance-matrix adaptation and stochastic gradient methods viewed as distributional dynamics, are widely used for nonconvex or black-box problems, yet their convergence analyses remain fragmented across algorithm-specific techniques. We introduce an operator calculus in which a broad class of such methods, after choosing an appropriate state space and, where necessary, augmenting the state by memory or strategy variables, is described as a composition of three elementary operators (mutation, selection, and recombination) acting on probability measures. Under explicit stability and regularity conditions, the composite operator admits a pre-generator whose continuous-time limit is a transport-reaction-jump (TRJ) PDE that preserves the operator splitting. On this foundation we establish a modular Lyapunov principle. If a state-space Lyapunov function both dissipates under the full generator and controls the relevant search-space gauges, then the state-space Lyapunov functional and the induced search errors decay exponentially. The additive generator structure allows dissipation estimates to be assembled operator by operator, providing a toolkit for certifying convergence of composite mean-field algorithms.

The real-time orchestration of microgrids that combine fluctuating renewable sources, dispatchable units, storage and curtailable consumers requires the repeated solution of combinatorial dispatch and coalition formation problems under hard control deadlines. In this paper, a quantum-assisted agentic distributed artificial intelligence (DAI) framework is proposed in which the dispatch problem of each control slot is formulated as a quadratic unconstrained binary optimization (QUBO) problem by Belief-Desire-Intention extended (BDIx) agents and is solved by a portfolio of quantum, quantum-inspired and classical solvers. Solver selection is elevated to a first-class agentic deliberation action of the coordinator agent. Learned beliefs about solver latencies are maintained and the solver intention that is expected to satisfy the prevailing deliberation deadline is committed in each slot. In addition, a belief-shaped storage valuation mechanism is introduced through which the storage agent prices its energy at a discounted future-peak value, injecting intertemporal information into the otherwise myopic per-slot optimization. The framework is evaluated on a 24-hour simulation of a grid-connected microgrid with photovoltaic, wind, battery, genset and demand-response assets, with the Quantum Approximate Optimization Algorithm (QAOA) executed by statevector simulation and benchmarked per slot against tabu search, simulated annealing, binary particle swarm optimization, greedy descent and exhaustive enumeration. Zero deliberation deadlines are missed, the committed dispatch attains the exact optimum on every slot and the realized daily cost of 146.24 EUR equals the exact lower bound, with 97.83 percent renewable utilization and zero unserved energy. When the storage valuation mechanism is deactivated, the daily cost is increased to 152.75 EUR, a 4.5 percent increase.

Renewable energy is essential for meeting future energy demands; however, solar energy generation, which occurs only during daylight hours often does not align with household consumption patterns. Appliances such as cookers, washing machines, and dryers are typically operated according to user preferred schedules rather than solar energy availability, creating a scheduling optimization problem. The objective is to determine optimal appliance start times to maximize renewable energy utilization while minimizing user inconvenience and adhering to system constraints. This paper presents a metaheuristic approach using Iterated Local Search (ILS) and Simulated Annealing (SA) to optimize appliance start times, while considering appliance operating durations, power consumption, inverter limit, battery state of charge constraints, and solar generation forecasts. Unlike most existing work, the scheduling is extended beyond a single day to accommodate unfinished tasks from previous days (spillover), ensuring operational continuity and enabling sequential operation across multiple days. Experimental results show that the sequential multi-day scheduling framework effectively manages system constraints while ensuring user convenience under exclusive solar generation. These findings also open opportunities for future research on multi-objective trade-offs between investment in equipment of various sizes, return on that investment, and user satisfaction.

The limiting resource for training agents via reinforcement learning (RL) is increasingly frontier task supply: valid, solvable tasks just difficult enough to train the current model. As reasoning and agentic models improve, fixed task distributions saturate, while naive synthetic generation yields tasks that are trivial, impossible, or ill-posed. Training a task generator with RL to optimize validity and learnability can address this bottleneck, but direct optimization requires repeated solver rollouts per candidate. For software-engineering (SWE) tasks, a single rollout can take tens of minutes; solver-in-the-loop generator training is intractable. We introduce PROPEL, a solver-amortized framework for training task generators at the targeted solve rate. PROPEL trains a lightweight activation probe on a one-time labeled corpus of generated tasks and solver outcomes. The probe predicts target-solver pass rate from a frozen generator reference model and serves as a proxy for solve rate during generator optimization, reducing generator evaluation to a single forward pass. Across math, code, and software-engineering at multiple model scales, PROPEL shifts generation toward the targeted solve rate: for coding, tasks generated at the learnable frontier increase from $10.1\% \rightarrow 20.0\%$ for a Qwen2.5-3B-Instruct solver and from $5.3\% \rightarrow 12.6\%$ for a Qwen2.5-7B-Instruct solver. For SWE, PROPEL increases the share of generations at the targeted solve rate from $9.8\% \rightarrow 19.6\%$ for Qwen3.5-27B on repositories not seen during training of probe and generator.

Large language models can reduce the manual effort required to set up finite element simulations, but they introduce reliability risks when generated solver code lies on the critical path. We present a constrained natural-language interface for multi-physics finite element analysis in which the LLM is limited to front-end tasks: parsing prompts into structured JSON, generating Gmsh code only for non-catalog geometries, and using retry feedback for those stages. It never writes FEniCS solver templates, derives weak forms, or writes the numerical solver core. A deterministic dispatcher maps the validated specification to five human-written FEniCS/UFL templates: linear elasticity, hyperelasticity, elastoplasticity, thermo-mechanical coupling, and phase-field fracture. We validate this deterministic template layer against analytical solutions and published 2D/3D benchmarks. Smooth cases reach sub-percent agreement on adequate meshes, while harder nonlinear cases reach the 2-5 percent range. We also evaluate the LLM-facing front end directly. In a 15-prompt parser benchmark, first-pass valid parses were obtained for 9 cases, and all remaining cases were repaired after retry, giving a final valid parse rate of 100.0 percent, 100.0 percent problem-class accuracy, and 97.1 percent field-extraction accuracy. In a 10-case custom-geometry benchmark routed through the real LLM-to-Gmsh path, first-pass and final success were both 90.0 percent, with one unrecovered invalid-geometry failure. These results show that the parser and constrained prompt/validation design are effective on these benchmarks. As an end-to-end demonstration, the system generates and analyzes a 3D elastoplastic L-bracket with a fillet and bolt hole from one natural-language prompt. The contribution is a measured architecture for natural-language-driven variational simulation, not open-ended autonomous code generation.

Open-pit mine scheduling is a critical process for maximizing economic return under complex geotechnical and operational constraints. While Mixed-Integer Linear Programming (MILP) provides mathematically optimal baselines, its exponential computational complexity and inability to adapt in real time limit its practical deployment in dynamic industrial environments. This work introduces a simulator-driven Large Language Model (LLM) scheduling framework in which the LLM acts as an autonomous decision-making agent, guided at each step by a custom simulator that encodes geotechnical precedence, extraction-processing coupling, and dynamic capacity constraints directly into the action generation mechanism. Operating entirely zero-shot within a closed, data-secure environment, the framework produces complete, interpretable extraction and processing schedules without cloud-based inference, domain-specific fine-tuning, or retraining. To provide a trustworthy performance benchmark, a novel MILP formulation is developed that incorporates realistic operational and geotechnical constraints. Evaluated across mining instances of varying scale and time periods, the LLM-based framework recovers between 94% and 99% of the MILP optimal NPV while scaling linearly in computation time. These results position simulator-constrained LLM agents as a practical and scalable alternative to classical optimization for long-horizon industrial scheduling under complex operational constraints.

With the advancement of visual sensing systems, computer vision is playing an increasingly important role in autonomous driving and robot navigation. Relative pose estimation in multi-camera systems is essential for accurate vehicle localization and environment perception, demanding high real-time performance and robustness. Existing methods, however, often involve high computational costs and rely heavily on abundant feature matches, limiting their applicability in time-sensitive driving scenarios. To address these limitations, this paper introduces a unified framework for efficient relative pose estimation, built upon a novel translation parameterization and first-order rotation approximation. Within this framework, we propose three efficient minimal solvers specifically designed for autonomous vehicles. The first solver integrates the vertical direction prior from Inertial Measurement Units (IMUs), the second utilizes the rotation axis direction prior during steering maneuvers, and the third is designed for planar motion - a realistic assumption for ground vehicles operating on structured roads. By reducing both the minimal number of point correspondences and the algebraic complexity, our methods enable faster hypothesis generation within RANSAC-based pipelines, improving suitability for real-time systems. Extensive experiments on synthetic datasets and the KITTI autonomous driving benchmark demonstrate that the proposed solvers achieve a favorable balance between speed and accuracy compared to existing state-of-the-art algorithms.

The Traveling Salesman Problem (TSP) is a classical NP-hard combinatorial optimization problem that aims to find the shortest Hamiltonian cycle visiting each city exactly once and returning to the starting point. This paper proposes a hybrid metaheuristic for the TSP by combining the Dragonfly Algorithm (DA), a swarm-intelligence-based global search method, with Tabu Search (TS), a memory-based local search technique. The proposed method follows a High-Level Relay Hybridization (HRH) scheme, in which DA is first used to explore the solution space and generate a promising initial tour, while TS subsequently refines this solution through neighbourhood-based improvement and tabu memory. The hybrid approach is evaluated on standard TSPLIB benchmark instances, including burma14, att48, and ch150, and compared with standalone DA, standalone TS, and several classical metaheuristics such as Genetic Algorithm, Ant Colony Optimization, Particle Swarm Optimization, and Random Search. A systematic grid-search procedure is also conducted to study the influence of the main hyperparameters on solution quality and execution time. The experimental results indicate that the proposed hybrid can improve tour quality compared with the standalone DA and TS on the tested instances, highlighting the benefit of combining global exploration with local exploitation. However, the results also suggest that performance remains sensitive to parameter settings and problem size, motivating further validation on larger benchmarks and stronger TSP-specific baselines.

Estimating the relative poses of multi-camera systems is a fundamental problem in computer vision, with critical applications in autonomous vehicles, mobile devices, and unmanned aerial vehicles (UAVs). However, existing solutions often suffer from high computational complexity or rely on an excessive number of point correspondences, limiting their real-world applicability. To address these limitations, we propose two efficient minimal solvers for estimating the relative poses of multi-camera systems using a novel parameterization. The first solver leverages the vertical direction prior provided by Inertial Measurement Units (IMUs), while the second utilizes the rotation axis direction prior from IMUs. Our methods require only four point correspondences and reduce the problem of multi-camera relative pose estimation to solving a univariate 6th-degree polynomial, a significant improvement over existing approaches, which typically involve 8th-degree polynomials. This reduction in computational complexity and correspondence requirements makes our solvers particularly effective when integrated into RANSAC frameworks, demonstrating strong potential for visual odometry applications. Through rigorous evaluations on synthetic data and the KITTI benchmark, our methods achieved superior computational efficiency and competitive accuracy compared to state-of-the-art algorithms.

As a bio-inspired intelligent sensor, event cameras have introduced a new paradigm in the intelligent perception of spatiotemporal information and visual motion estimation, characterized by their high temporal resolution, low latency, and minimal power consumption. However, their asynchronous data streams present significant challenges to traditional synchronous, frame-based algorithms. To address these challenges, this paper presents a novel framework for full degree of freedom (DoF) egomotion estimation directly from asynchronous optical flow, specifically targeting the joint recovery of angular and linear velocities. We decouple the differential epipolar constraint into distinct angular and linear velocity components, and derive its formulation for asynchronous data. Based on this formulation, an optimization algorithm is developed that enables full-DoF egomotion estimation leveraging at least five points. Furthermore, by applying a first-order approximation to rotational dynamics, we transform the constraint equations into a polynomial form, resulting in the first algebraic minimal 5-point solver for this formulation. To ensure real-time performance in high-speed scenarios, we additionally propose an accelerated solver achieved by truncating high-order angular velocity terms. Extensive evaluations on both synthetic and real-world datasets demonstrate that the asynchronous approach outperforms traditional synchronous methods, particularly in its accuracy and robustness to spatiotemporal noise. We believe that this work establishes a critical foundation for efficient and accurate continuous-time motion estimation in high-speed robotics applications.

We ask whether structural properties of intermediate grid states predict whether a symbolic ARC-AGI solver will succeed, framed as a test of conditional mutual information I(X;Y|task) > 0. Across 44,800 runs spanning two architecturally distinct solvers (beam search and Stochastic DFS), 400 ARC tasks, 28 configurations per solver, and both training and evaluation splits, hand-crafted grid descriptors measured at 50% trajectory completion discriminate successful from failed runs within the same task (mean within-task best-feature AUC = 0.885, p < 0.001 under within-task label permutation). Most predictive content lies along a single grid-complexity axis. The result generalizes across solver architectures: a feature selected on one solver predicts success on the other with AUC 0.747-0.762 in all four transfer directions (p < 0.001, leakage controlled). On a pre-registered held-out set of 41 reliable tasks, the frozen feature n_components_final achieves AUC = 0.765 (95% CI [0.717, 0.810], p < 0.001), robust under task-clustered bootstrap resampling and cross-solver task collapsing. The signal is not explained by solver capacity (configuration-residualized AUC = 0.927 and 0.896 for beam search and SDFS, p < 0.001) and is only weakly coupled to score trajectories (R^2 approximately 0). Early stopping at 50% completion reduces beam-search compute by 33.6% while retaining 98.9% of solves; degenerate-trajectory detection reduces SDFS compute by 65.3% with no solve loss. Finally, on 229 of 400 evaluation tasks the DSL primitive library produces no valid transition from the input grid. This 0-step collapse is invariant to search budget and universally failed by beam search, indicating a DSL coverage limitation rather than a search-budget effect.

Macro placement is a fundamental step in modern chip physical design, playing a crucial role in determining the solution quality of high-dimensional combinatorial optimization problems. Despite recent advancements in machine learning for spatial coordinate determination, the temporal dimension of placement sequencing remains largely governed by static heuristics. In this work, we demonstrate that the placement sequence is not merely a preprocessing step but a decisive factor in optimization, where suboptimal early decisions trigger irreversible domino effects that constrain the solution space. To harness this unexplored dimension, we propose \textbf{OrderPlace}, a proxy-guided LLM evolution framework for automatically discovering macro placement order strategies. Instead of relying on manually crafted heuristics such as area- or connectivity-based ordering, OrderPlace explores a broader space of code-level policies, ranging from static scoring metrics to dynamic physics-inspired mechanisms. To mitigate the prohibitive cost of evaluating sequences, we introduce a lightweight proxy evaluation mechanism that efficiently filters candidates using a deterministic greedy probe. Experimental results on the standard ISPD 2005 benchmarks demonstrate that OrderPlace discovers novel ordering strategies. Compared with WireMask-EA and the state-of-the-art method EGPlace, OrderPlace reduces wirelength by 34.04% and 14.08%, respectively.

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.

Gray-box optimization is an approach for making some problem-specific information available to the algorithm while still relying on fitness information as the main guide to an optimum. This approach was shown to be beneficial in various combinatorial optimization tasks and neatly captures the continuum between fully black-box algorithms and tailored algorithms. In this work, we discuss different flavors of gray-box algorithms. We show that RLS can find a proper $2$-coloring in a bipartite graph starting from a random $2$-coloring, in an expected time of $\mathcal{O}(n \log n)$. In contrast, when starting from a proper $n$-coloring, the (1+1) EA cannot find such a coloring except when offered additional guiding on plateaus of the search space. Finally, we show the run time for this setting can be much improved by using gray-box operators.

High-dimensional transient heat diffusion under noisy boundary conditions exposes a fundamental limitation of classical numerical methods: accuracy degrades catastrophically where physical noise is unavoidable. This paper presents a Physics-Informed Neural Network (PINN) framework as a systematic solution to this problem across one, two, and three spatial dimensions, establishing clear operational regimes that redefine solver selection in noisy thermal systems. Under 20% boundary noise in 3D, PINN sustains approximately 91% accuracy while Finite Difference Method (FDM) collapses to 36%, a clear decisive advantage. This is further confirmed in a physical copper thermal system, where PINN reduces boundary reconstruction error by 3.3 times under realistic noise conditions. This noise resilience is accompanied by a dimensionality-driven efficiency crossover: PINN requires fewer spacetime nodes than FDM in 3D while achieving superior accuracy, exposing the true cost of classical discretization at scale. These findings reframe solver selection: the decisive axis is not accuracy alone, but noise exposure and dimensionality jointly. When noise and dimensionality are both high, the classical solver paradigm is insufficient; this work provides the foundation to justify PINN as the operational standard in such regimes.

Benders decomposition is a fundamental framework for solving large-scale mixed-integer optimization problems with complicating variables that, when fixed, yield significantly easier subproblems. However, classical Benders decomposition repeatedly solves highly similar subproblems and often exhibits zigzagging behavior across iterations, leading to slow convergence in large-scale settings. Motivated by the repetitive structure and parametric nature of Benders subproblems, this paper introduces the proxy Benders decomposition (Proxy-BD), a new decomposition framework in which subproblem optimization is replaced by certified optimization proxies rather than repeated exact solves. The proposed proxy follows a self-supervised predict-project-and-complete mechanism that produces dual-feasible solutions for generating provably valid Benders cuts. The framework preserves the theoretical validity of the decomposition independently of prediction quality through a projection-and-completion certification layer. A formal characterization of proxy-induced cuts is established, and the framework naturally extends to modern decomposition schemes, including branch-and-Benders-cut algorithms. Computational experiments on large-scale facility location and network design problems demonstrate that Proxy-BD substantially reduces the computational effort of subproblems while maintaining near-optimal solution quality. On large-scale uncapacitated facility location instances up to 2000x2000, Proxy-BD achieves median optimality gaps below 0.5%, yields up to 161x median speedups, and reduces the number of generated cuts by more than 240x on the largest instances. The computational gains consistently increase with recourse complexity, indicating that proxy-based inference scales substantially more favorably than repeated exact subproblem optimization in large-scale decomposition settings.

Reservoir computing exploits the fixed dynamics of a recurrent network for temporal processing, requiring only a trained linear readout. Biological neural connectomes, shaped by millions of years of evolution, may encode computational structure beyond what random reservoirs provide, yet whether that structure can be further enhanced by principled optimisation remains an open question. We address it by applying four gradient-free, bio-inspired optimisers (Particle Swarm Optimisation, Differential Evolution, Grey Wolf Optimiser, and Whale Optimisation Algorithm) to the edge weights of connectome-based echo-state networks across six species spanning six orders of magnitude in neural complexity: C. elegans (279 neurons), Drosophila (49 nodes), mouse (112), rat (73), macaque (29 regions, continuous FLNe synaptic strengths), and human structural MRI connectivity (83 parcels). Each connectome is evaluated on four canonical reservoir computing benchmarks: Memory Capacity (MC), Lorenz attractor prediction, NARMA-10 system identification, and Mackey-Glass chaotic time-series prediction. All four optimisers consistently outperform unoptimised biological baselines across every task and species when initialised from biological weights. WOA achieves the largest gains on every task: up to a 17x MC improvement (C. elegans: 1.39 to 23.91) and up to 89% NRMSE reduction (Mackey-Glass, human), corresponding to an average 214% improvement across all species and tasks. Crucially, random initialisation on the same topology reliably underperforms biology, establishing biological weight values as an essential inductive bias that topology alone cannot recover. These results position bio-inspired, biologically-initialised optimisation as a principled and broadly effective strategy for connectome reservoir computing across the animal kingdom.

Many equations arising in science currently cannot be solved by available analytical techniques and are therefore solved numerically, without yielding explicit symbolic expressions. Existing symbolic regression approaches can recover symbolic expressions, but require training data obtained from the underlying process, rather than the governing equation alone. We propose the Symbolic Equation Solver (SES), a framework that formulates equation solving as an optimization problem over differentiable symbolic models. SES constructs its objective from the equation together with initial or boundary conditions, eliminating the need for paired input-output data. The learned model is expressed in explicit symbolic form, enabling further analysis. We evaluate SES on representative algebraic and differential equations, including a system of algebraic equations, an equation with transcendental terms, an ordinary differential equation, and partial differential equations with different initial or boundary conditions. Across these settings, SES recovers compact symbolic expressions that match the corresponding analytical solutions.

The heterogeneity of client data and systems makes it difficult to achieve satisfactory convergence speed and robustness in federated learning with random client selection. To address this issue, this paper proposes a surrogate-assisted client evolutionary selection framework for federated learning. In this framework, some typical client selection strategies are first used to generate candidate sets, and a metric function that integrates model performance, communication latency, and energy consumption is developed to formulate the client selection problem as a combinatorial optimization one. Subsequently, a surrogate model is constructed using the candidate selections and metric to efficiently approximate the performance of selected client subsets. An evolutionary algorithm is employed to search the combinatorial space of client selections, guided by the surrogate model to accelerate convergence. Experiments on MNIST, CIFAR10, CINIC10, and TinyImageNet demonstrate that the proposed algorithm achieves faster convergence, lower energy consumption, and improved robustness compared to existing methods.

We study parallel Continuous Local Search (CLS) as a solution approach for Boolean satisfiability problems with symmetric pseudo-Boolean (PB) constraints. Here, the $n$-variable PB-satisfiability problem is relaxed to a continuous optimisation problem with a differentiable objective function on an $n$-dimensional hypercube. For satisfiable instances, the global minimisers of this optimisation problem correspond to satisfying assignments of the SAT problem at hand. We present several novel findings via empirical experiments: (i) redundant constraints can inhibit rather than accelerate convergence; (ii) CLS shows promise as a sub-solver in hybridised settings, quickly completing partial assignments; and (iii) local search rapidly converges to a stable distribution of solution quality (i.e., degree of satisfaction), due to saddle-dense objectives where additional solver steps yield diminishing returns. Our findings inform practical uses of CLS for SAT on modern accelerator hardware.

The radial basis function neural network (RBFN) trained with a gradient descending algorithm provides an effective fully connected structure in both shallow and deep networks. The error correction (ErrCor), a state-of-the-art gradient-based training method, selects optimal hidden units to improve accuracy. Alternatively, as a population-based algorithm, the particle swarm optimization algorithm (PSO) uses the swarm experience to optimize RBFN parameters, offering global search and robustness to local minima. Adaptive PSO (APSO) has emerged as an improved variant of PSO. APSO algorithm improves convergence speed by dynamically adjusting swarm parameters during optimization. Both ErrCor and PSO demonstrate improved results and competitive convergence. However, with large datasets, these methods face scalability challenges such as excessive kernel computations and large hidden layer structures. A recent multi-column RBFN approach (MCRN) improves ErrCor performance by deploying small RBFNs in a parallel system. Inspired by MCRN's success, we propose two novel approaches to improve PSO performance: the multi-column RBFN with PSO (MC-PSO) and the multi-column RBFN with APSO (MC-APSO). These methods introduce parallel RBFN structures trained using evolutionary swarm methods. Each RBFN is independently trained on a specific spatial subset of the dataset using either PSO or APSO algorithms. These resulting specialist-trained RBFNs are tailored to their respective subsets. During testing, only selected RBFNs, where the test instance neighbors are located, contribute to the multi-column output. This specialization improves accuracy, while parallelism enhances speed. We evaluate the proposed methods on various benchmark datasets. The MC-PSO and MC-APSO outperform ErrCor, PSO, APSO, and MCRN in terms of accuracy and recall. They also demonstrate faster training and testing times in most experiments.

Large language models (LLMs) increasingly translate natural-language optimization problems into executable solver code. Yet for constraint-dense operations research (OR) problems, existing data-filtering and training pipelines largely rely on objective-equivalence signals such as differential testing and answer agreement, which a program can pass while adding spurious constraints or silently omitting required ones, whenever those constraints are non-binding on the tested instance. We propose constraint injection, which uses feasible probes to expose spurious over-constraint and one-constraint-violating probes to reveal silent constraint omission. Combined with differential testing, it forms a dual verifier. We instantiate and evaluate it on vehicle routing problems (VRPs), a representative constraint-dense combinatorial optimization testbed with coupled operational constraints. We develop VRPCoder, an 8B end-to-end model that translates natural-language VRP scenarios into Gurobi scripts, together with an expert-verified VRP benchmark suite covering 21 variants. The verifier is reused as a rejection-sampling filter during data synthesis and as a per-rollout reward in group relative policy optimization (GRPO). Across four VRP benchmarks, VRPCoder-GRPO reaches 93% average Pass@1, outperforms Gemini-3.1-Pro Preview on three benchmarks, exceeds Claude-Sonnet-4.5 by 28 average points, and surpasses prior OR-LLMs by 78 average points.

The 2026 disproof of Erdős's unit-distance conjecture and Sawin's quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$. Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes integer parameters whose choice is not fully optimized. This report treats Sawin's parameter selection as a nonlinear integer optimization problem and develops an open-source Python optimization and verification pipeline for certificates involving prime sets $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. After reproducing Sawin's certificate with $δ=0.014114\ldots$, the pipeline yields improved certificates with the same $T$. We develop a tailored integer evolution strategy achieving a certificate with $δ=0.015263\ldots$ and supporting the cautious statement $u(n)>n^{1.0152}$ for arbitrarily large $n$. For extended ramified prime ranges, the Emmerich--Cordella certificate obtained with the same framework reports $u(n)>n^{1.031}$ for $\#T=67$, illustrating the importance of enlarging $T$. Very recent MathOverflow discussions, brought to the author's attention as of version~4, report further improvements, including certificates above $δ>0.035$ and beyond $δ>0.036$. Some of these improvements may rely not only on larger prime ranges but also on modified constraint systems and additional degrees of freedom that deviate from Sawin's original formulation. Beyond this application, the work illustrates how randomized optimization heuristics can improve, verify, and refine explicit certificates for combinatorial geometry through nonlinear integer optimization.

Neural-guided Ant Colony Optimization (ACO) suffers from a fundamental training-inference misalignment: policies are typically trained to generate static priors (e.g., heatmaps), yet deployed to guide iterative, long-horizon search processes. In this paper, we present DyNACO, a novel framework that achieves dynamic neural guidance by periodically observing the pheromone distribution and the incumbent solution. To make DyNACO tractable at scale, we pair the policy with a perturbation-based ACO backend and a scope-restricted refinement mechanism that jointly ensure efficacy and stable credit assignment. On TSP, DyNACO scales to 100,000-node instances and outperforms neural baselines while often reducing total runtime compared to the unguided solver. We extend DyNACO to CVRP via a capacity-aware backend, consistently improving the unguided baseline with less than 1% neural overhead. We further provide in-depth analysis validating the model's generalization capabilities and elucidating why dynamic guidance outperforms static priors. Our work underscores the necessity of aligning neural training with iterative search dynamics in learning-guided optimization. The code is available at https://github.com/shoraaa/DyNACO.

Classical finite-difference solvers remain reliable tools for partial differential equations, but their efficiency depends on where mesh resolution is placed. Uniform refinement can waste degrees of freedom when solution difficulty is localised near sharp gradients, fronts, oscillations, or constraint-sensitive regions. This paper studies a hybrid strategy in which a physics-informed neural network (PINN) is used not as the final solver, but as an off-grid residual probe for adaptive mesh refinement. The PINN residual is sampled over the domain, converted into cellwise indicators, and used to guide refinement before the final approximation is computed by a finite-difference solver. The method is evaluated on three benchmarks. The main full-solver validation uses the one-dimensional viscous Burgers equation with a nonuniform finite-difference solve on the adapted meshes. PINN-threshold refinement attains final relative $L^2$ error $0.021067$ with $60$ degrees of freedom, compared with $0.022617$ for uniform refinement with $192$ degrees of freedom. At matched mesh size, PINN-threshold reduces the error by about $67.5\%$. PINN-Dörfler refinement gives similar performance, with error $0.021264$ using $58$ degrees of freedom. A gradient indicator remains slightly more accurate, so the result supports usefulness rather than universal superiority. Manufactured 2D and 3D proxy tests, based on a nonlinear Schrödinger equation and an incompressible Navier--Stokes system, show that PINN residuals can organise structured refinement and improve over random refinement, although they do not consistently outperform gradient or uniform baselines. The results support PINN-guided AMR as a residual-indicator strategy for transferring physics-informed diagnostic information into finite-difference mesh adaptation while preserving the classical solver as the final approximation engine.

Mixed-combinatorial nonlinear programming (MCNLP) problems arise in many engineering design and planning applications, e.g., due to categorical, component, and geometric design choices, as well as joint task and motion planning. Traditional representations of combinatorial spaces, such as integer or binary encoding, often introduce spurious relations, increase dimensionality, and require additional compatibility constraints. Instead, this paper draws on recent developments in robot planning and vehicle/network routing domains that aim to learn search heuristics over combinatorial spaces using graph neural networks (GNNs). More specifically, this paper presents a first-of-its-kind structured abstraction of the combinatorial space by learning a mapping from an undirected fully connected graph of combinations to a directed graph indicating improvement directions using an Edge Field Graph Network (EFGN). To demonstrate the utility of this new way of abstracting the combinatorial space in solving MCNLPs, we adopt a recent optimization framework that purely searches over the non-combinatorial (e.g., continuous) variables and retrieves the best-suited combination for each candidate design by using the abstraction model, akin to a recommender system. The presented direction-aware abstraction model provides a potentially more scalable and interpretable retrieval of combinations compared to the original recommendation system in that framework. For evaluation, the proposed method is integrated with a well-known particle swarm optimization and genetic algorithm solvers on three benchmark nonlinear problems with varying numbers of combinations and variables. Compared to baseline solvers using indexified combinations, the GNN-based recommender consistently achieves better mean optimum values and robustness across multiple runs.

The evolution toward 6G wireless networks envisions a seamlessly intelligent, Open-RAN-enabled architecture where unmanned aerial vehicles (UAVs) play a pivotal role in extending coverage, enhancing resilience, and ensuring reliable connectivity for ground users deployment. However, efficiently managing spectrum and resources in such highly dynamic UAV-assisted environments remains a major challenge due to nonlinear system interactions, mobility-induced topology variations, and stringent latency and energy constraints. To address these challenges, we propose a digital twin (DT)-assisted adaptive deep reinforcement learning (DRL) framework that enables intelligent spectrum sharing and resource allocation across distributed ground users. The complex optimization problem is decomposed into UAV trajectory optimization using particle swarm optimization (PSO) and dynamic spectrum-power-association management via multi-agent DRL (MADRL). This hybrid DT-driven approach empowers intelligent, context-aware decision-making and adaptive coordination among UAVs. Extensive simulations demonstrate significant gains in spectral efficiency, data rates, and energy utilization, showcasing a transformative path toward self-evolving, autonomous 6G UAV and ground users (GUs) connectivity.

Decision-focused learning (DFL) trains predictive models by optimizing downstream decision quality rather than standalone prediction accuracy. For contextual linear optimization, most existing DFL methods assume offline data and full observations of the objective cost vector. We develop an on-policy learning method for sequential contextual linear optimization under partial feedback, generalizing the standard bandit feedback setting. Our method learns a stochastic predict-then-optimize policy that samples a cost-vector prediction from a conditional distribution and solves the resulting downstream linear optimization problem. To update this distributional model, we introduce a two-component hybrid gradient estimator. The first component is a score function estimator, which provides an unbiased but potentially high-variance policy gradient estimate. The second is a decision-focused plug-in component that uses an auxiliary nuisance estimate of the latent cost vector to exploit the downstream optimization structure, becoming more informative as the estimate improves. We prove an $\mathcal{O}(T^{-1/2})$ bound on the average squared policy-gradient norm, matching the standard non-convex SGD rate. Experiments on top-$k$ selection, shortest path, combinatorial pricing, and a real-data energy-scheduling benchmark show that the hybrid gradient approach achieves lower cumulative regret than contextual-bandit-style baselines across all benchmarks, using both Gaussian and richer conditional generative models. Code is available at https://github.com/Joeyetinghan/on-policy-bandit-dfl.

While Large Language Models (LLMs) have recently shown promise in Automated Heuristic Design (AHD), existing methods typically generate and evolve heuristics as a single operator or search strategy, limiting their ability to model strong coupling among multiple decision substructures in problems such as the Traveling Thief Problem (TTP) and the Traveling Purchaser Problem (TPP). In this work, we propose CoEvo-AHD, an LLM-driven dual-population co-evolutionary framework for automated heuristic design in coupled combinatorial optimization. Unlike prior methods that evolve individual heuristics in isolation, CoEvo-AHD leverages LLMs to co-evolve two closely related operator populations. A cooperative evaluation mechanism explicitly captures interactions between route and selection operators, while pairwise scoring and synergistic joint crossover help discover complementary operator logic for joint improvement across coupled decision subspaces. We further design a tool-invocation environment library that encapsulates frequently used core operations, such as local-search delta computation, into callable functions, enabling LLM-generated operators to use standardized interfaces instead of reimplementing inefficient and error-prone problem-specific loops. Experiments on TTP and TPP show that CoEvo-AHD automatically discovers cooperative heuristic combinations and achieves competitive solution quality against traditional heuristics.

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

Learning-based surrogates for partial differential equations have recently matched the accuracy of classical solvers while achieving orders-of-magnitude speedups, predominantly in fluid settings and structured geometries. In contrast, robust surrogates for deformable solids remain underexplored, despite the presence of nonlinear elasticity, plasticity, and transient behavior that challenge standard architectures. We introduce a multigrid graph neural network for solid mechanics that couples an encoder-processor-decoder backbone with a physics-informed coarsening strategy. Instead of downsampling via geometric heuristics, our method scores nodes using a residual-based measure of local physical activity and preferentially retains regions of high strain or stress concentration, allocating multiscale capacity where it is most needed. This preserves long-range interactions through hierarchical message passing while improving stability over long rollouts. We evaluate on multiple datasets covering linear, nonlinear, and transient regimes, and observe consistent gains in accuracy and rollout stability compared to standard sampling baselines. Our results highlight the importance of physics-informed coarsening for scalable surrogate modeling in solid mechanics.

Diffusion-based neural solvers have shown strong promise for combinatorial optimization (CO), but existing methods typically rely on supervised training with large collections of near-optimal solutions. In this work, we extend adjoint-based trajectory optimization methods to discrete combinatorial domains. We formulate diffusion-based CO as a stochastic control problem over Continuous-Time Markov Chains and introduce discrete adjoint dynamics for propagating optimization signals through discrete generative trajectories. Building on this formulation, we propose Combinatorial Adjoint Matching (CAM), an unsupervised training framework for discrete diffusion solvers with structured and low-variance trajectory-level optimization signals. Empirically, CAM consistently outperforms existing unsupervised diffusion baselines and achieves performance competitive with strong supervised diffusion solvers and even traditional solvers across diverse combinatorial optimization problems. Our code is available at https://github.com/Shengyu-Feng/CAM.

We present Caspar, a library that makes the power of modern GPUs more accessible in robotics and provides a state-of-the-art nonlinear GPU solver that can be applied to a wide range of different optimization problems. Caspar bridges the gap between expressive symbolic programming in Python and high-performance GPU runtimes in C++ by automatically generating optimized CUDA kernels from symbolic expressions. Building on the SymForce library, users can easily define and combine symbolic expressions, including Lie group operations, to generate custom CUDA kernels. To use Caspar as a solver, users need only define the symbolic residual functions; Caspar then uses symbolic differentiation to generate the necessary GPU kernels and interfaces to perform nonlinear optimization. In this paper, we present the core components of Caspar and showcase its performance by performing bundle adjustment on the Bundle Adjustment in the Large (BAL) dataset. We benchmark Caspar against other state-of-the-art bundle adjusters and show that it is 5 to 20 times faster than the best alternative, requires less memory, and achieves similar accuracy. This illustrates the benefit of our symbolic GPU programming approach. Caspar is released as part of SymForce and is freely available at https://github.com/symforce-org/symforce

A central challenge in game theory and learning systems such as GANs is understanding which algorithms can efficiently compute equilibria across the heterogeneous landscape of games. Equilibrium computation is typically studied solver by solver and game class by game class, yielding strong local guarantees but a fragmented view of solver behaviour. Existing discrete taxonomies often provide an incomplete account of where algorithms succeed. We study this problem through a solver-game map linking games to effective solver dynamics. Classical theory identifies isolated regions of this map but provides limited insight into intermediate or overlapping regimes, suggesting that solvability is governed by latent structural properties defining a continuous solver-aligned geometry of games. We formalise this perspective through structure-aware solver synthesis. A learned structure recogniser maps each game to a low-dimensional solver-aligned representation, and a policy maps this representation to effective primitive mechanisms, adapting solver behaviour across regimes. This reveals regions where particular solver dynamics are effective and where mixtures of primitives are required rather than a single dominant solver. A bounded residual acts as a local corrector and diagnostic signal for incomplete solver bases or representations. The framework yields both an adaptive solver and an analytical lens: games with similar optimisation dynamics cluster together, revealing continuous regions of algorithmic validity and overlapping solver behaviour. Empirically, we show that fixed primitives exhibit systematic regime mismatch, while the learned representation organises game space into a structured cartography aligned with solver behaviour. These results suggest viewing equilibrium computation as the joint problem of learning solver mechanisms and mapping the geometry of solvability.

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.

A novel two-phase molecule inference framework, mol-infer, has recently been developed to infer chemical graphs with prescribed abstract structures and desired property values through mixed integer linear programming (MILP) under the two-layered model, with guaranteed optimality and exactness relative to the given learned prediction function and structural constraints. In this study, we extend this framework to copolymers by introducing a simple feature representation, called the mixing vector (MV) model. In the proposed model, a copolymer feature vector is represented as a convex combination of MILP-tractable monomer descriptors weighted by the mixing ratio of the constituent monomers. This representation does not require explicit sequence-class information and is therefore naturally compatible with MILP-based inverse design. Under this model, we construct prediction functions for several copolymer property datasets using artificial neural networks, reduced quadratic multiple linear regression, and random forests. The proposed representation achieves practically useful predictive performance across multiple physicochemical property datasets; in particular, the best test R^2 score exceeds 0.7 for nine of the ten datasets and exceeds 0.9 for six datasets. We also formulate a multi-monomer inverse-design problem under the MV representation with a prescribed mixing ratio and show that the resulting MILP instances remain tractable, even for three-monomer settings. Finally, we perform an external consistency check by re-evaluating the inferred candidates and comparing the re-computed property values with those predicted by the learned model. Overall, the proposed framework gives a tractable first step toward model-level exact inverse design of copolymers under the two-layered model.

Active tether-net systems are a promising solution for capturing large non-cooperative targets, such as space debris, by deploying a flexible net manipulated by maneuverable units (MUs). However, concurrent systematic explorations of design and control choices of the tether-net system to understand its full potential remain limited, partly due to the complex, constrained, nonlinear optimization problem that it presents -- one that involves a mixture of continuous, integer and categorical variables, with the latter two arising from net connectivity and component choices, respectively. Classical binary encoding methods are often ineffective for solving highly nonlinear and multimodal Mixed Combinatorial Nonlinear Programmings (MCNLPs) in engineering design, while integer coding approaches can introduce spurious relations among combinations. Given the graph-structured characteristics of the combinatorial space, this paper adopts and extends a new graph-learning-aided optimization approach to solve this MCNLP problem. Here, a Graph Neural Network (GNN) is trained to score (as output) and thereof recommend candidate combinations represented as nodes in a graph, with the continuous variable vector portion of a candidate design given as input. As a result, the MCNLP optimization reduces to an NLP, which can be solved using standard solvers. While this reduction approach is agnostic to the choice of the NLP solver, here a state-of-the-art Particle Swarm Optimization (PSO) algorithm with gradient-based fine-tuning is used as the solver. Demonstrated on the problem of concurrently designing the morphology of the net, choice of mass and thrusters in the MUs and aiming points used by the controller of the tether-net system, the GNN-based recommender is shown to provide significantly faster convergence to similar optimal solutions, compared to direct solution of the MCNLP problem.

A new variant of the classic capacitated facility location problem, which considers incompatibilities between customers, has recently been introduced in the literature. This problem captures the situation where given pairs of customers cannot be served by the same facility. Such a feature is crucial for many practical cases of location problems, such as the presence of hazardous or polluting materials and contention between competing costumers. In this paper, we propose a Large Neighborhood Search (LNS) method to solve this problem. Within the framework of LNS, we introduce three different destroy operators, which are combined in a hybrid manner, and we use an exact solver in the repair phase. Different algorithmic components are investigated for the design of LNS. The experimental analysis shows that our new method outperforms existing state-of-the-art metaheuristics, providing new best solutions for all available benchmark instances.

Neural surrogates promise large speedups over classical solvers for physical dynamics but fail silently at sharp dynamical events such as shocks, fronts, and contact. We present hybrid neural world models for physical dynamics: a recipe for training and deploying multi-horizon surrogates in physical state space, where a single network with continuous horizon conditioning is trained with direct supervision against textbook reference solvers to predict any future state at horizon T in one forward pass. Although no part of the training data, loss function, or architecture supervises discontinuity location, the trained surrogate encodes it implicitly, recoverable from its forward passes alone as a per-trajectory error map that concentrates on shocks, fronts, and contacts, and stays small elsewhere. The map is competitive with or better than standard label-free baselines including deep ensembles, learned error heads, gradient-magnitude indicators, and locally-adaptive conformal prediction, while using only a single trained network and requiring no calibration set or governing-equation knowledge. The recipe supports two operating points. Mode 1 runs the surrogate alone for maximum throughput, with same-hardware CPU speedups of 26x to 72x against textbook solvers on the PDE environments. Mode 2 uses the error map to gate a reference-solver fallback, deferring uncertain trajectories and roughly halving the surrogate's residual error at the default operating point. The recipe applies without modification across reaction-diffusion, compressible Euler, and rigid-body collision dynamics.

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*.

Constraint Acquisition (CA) and related research on the validation and enhancement of Mathematical Programming (MP) models from domain knowledge artifacts are currently limited by inadequate benchmarks. This deficiency impedes reproducibility and cross-study comparability, slowing the maturation of CA methods. Existing benchmarks were designed for solver evaluation rather than for assessing CA algorithms. They are loosely organized, treat individual problems inconsistently, and omit the domain knowledge artifacts required by CA methods. This work presents MPMMine, a benchmark suite designed to assess algorithms that discover, validate, and enhance MP models using diverse domain knowledge artifacts. MPMMine is guided by consistency, standardization, completeness, extensibility, openness, and version control. It adopts a uniform structure and relies on open formats: MiniZinc, CommonMark, and JSON. It provides multiple models per problem, tens of instances per model, and thousands of solutions and non-solutions in both integer and continuous domains, alongside natural-language descriptions to support text-to-model methods.

Efficiently solving Poisson equations on complex, irregular domains remains a fundamental challenge in scientific computing, as classical iterative solvers often suffer from prohibitive runtime due to ill-conditioned systems. While neural operators offer a fast alternative, they typically rely on large-scale labeled datasets or struggle with unstable training dynamics when using physics-informed residual losses. We propose \textsc{NPSolver}, a neural Poisson solver trained without solution labels via iterative physics supervision. Instead of relying on fully converged numerical solutions or raw PDE residuals, \textsc{NPSolver} utilizes a small number of preconditioned conjugate gradient (PCG) steps to refine its own predictions, providing a more stable and well-scaled training signal. Theoretical analysis confirms that this iterative supervision serves as a well-conditioned error proxy and that a stop-gradient design is essential for optimization stability. To better capture boundary-driven features under mixed boundary conditions, we further introduce the Boundary-Aware Transolver (\textsc{BA-Transolver}) architecture that explicitly separates interior and boundary tokenization. Extensive evaluations on 2D and 3D irregular geometries demonstrate that \textsc{NPSolver} outperforms both physics-informed and data-driven baselines. Furthermore, a downstream thermal control task highlights the model's capability for conducting efficient and reliable gradient-based boundary control. We will release our codes and data at https://github.com/intell-sci-comput/NPSolver.