mixed-integer programming

Non-binary integer variables in mixed-integer linear programming (MILP) impede end-to-end learning methods from guaranteeing solution feasibility—a fundamental challenge unaddressed by existing learned solvers. Method: We propose the first solver-free, reinforcement learning–driven framework for MILP, built upon Proximal Policy Optimization (PPO) and graph neural networks (GNNs). Our end-to-end architecture jointly models constraint satisfaction for both binary and non-binary integer variables. We introduce a novel branch-and-assignment joint action space and a self-supervised feasibility reward mechanism to enable progressive optimization—from the first feasible solution to near-optimal solutions. Results: On standard benchmarks, our method achieves 100% feasibility rate, attains an average objective value at 99.2% of the optimal, and solves problems 3.8× faster than traditional branch-and-bound—outperforming all prior learning-based solvers in both solution quality and efficiency.

To address the low computational efficiency of large-scale mixed-integer linear programming (MILP) solvers and the poor scalability of end-to-end learning approaches, this paper proposes a novel paradigm centered on “learning equivalence-preserving model reduction.” We introduce a preference-driven model reduction learning framework: it models relative performance preferences among fully reduced MILP formulations, employs an attention mechanism to capture pairwise preference relations, and incorporates a SetCover-based pruning strategy to efficiently control label space size. Experiments on real-world MILP instances demonstrate that our method improves solution accuracy by nearly 20% over state-of-the-art model reduction techniques and achieves 2–4 orders-of-magnitude speedup compared to the commercial solver Gurobi. The approach thus significantly enhances both accuracy and efficiency in solving large-scale MILP problems.

This work addresses mixed-integer linear programming (MILP) and stochastic optimization problems by proposing a probabilistic solution framework grounded in the Boltzmann distribution. The original problem is reformulated as a Monte Carlo optimization task—sampling from truncated multivariate exponential and Gaussian distributions over the feasible constraint set—and solved efficiently via the Kent–Davis sampling algorithm. Unlike conventional deterministic solvers, this approach avoids strong structural assumptions on the problem, thereby enhancing scalability and stochastic exploration capability. Experiments on portfolio optimization and the canonical stochastic farmer problem demonstrate that the method achieves solution quality comparable to state-of-the-art commercial solvers (e.g., Gurobi) on medium-scale instances, while exhibiting superior robustness to high-dimensional, non-convex, or black-box constraints. The framework introduces a novel probabilistic modeling and optimization paradigm for MILP, bridging statistical sampling theory with discrete and stochastic decision-making.

Traditional mixed-integer linear programming (MILP) modeling struggles to accommodate the diversity of complex combinatorial optimization problems, resulting in cumbersome formulations and limited solver compatibility. This work proposes OptiDSL, a novel domain-specific language (DSL)-centric modeling paradigm that leverages large language models to automatically translate natural language descriptions into structured DSL representations, thereby decoupling modeling from solving. By moving beyond MILP constraints, the framework enables flexible integration of diverse specialized, heuristic, and learning-based solvers. Evaluated across 44 combinatorial optimization problem classes, OptiDSL achieves a 51.66% improvement in modeling accuracy and reduces modeling time by 91.71% compared to conventional MILP approaches, while further boosting solution accuracy by 23.09% over existing benchmarks.

The absence of a universal, domain-agnostic dynamic programming (DP) modeling paradigm hinders systematic DP application to combinatorial optimization. Method: This paper introduces Domain-Independent Dynamic Programming (DIDP), a paradigm that decouples problem modeling from solving. We design DyPDL—a formal language for specifying DP models—and develop CAASDy, a general-purpose solver that unifies classical DP, A* search, and cost-algebraic state-space search within a verifiable DP framework for the first time. CAASDy supports interoperable interfaces with MIP and CP models, enabling fair empirical comparisons. Results: Experiments across multiple standard combinatorial optimization benchmarks demonstrate that CAASDy significantly outperforms leading commercial MIP and CP solvers. These results validate DIDP’s triple innovation: modeling generality, solving efficacy, and theoretical verifiability.

This study addresses the challenges of managing competing ideas and controlling long-cycle experimental trajectories in automated research for mixed-integer programming (MIP) by proposing the AutoMIP framework. This method is the first to integrate a persistent idea pool with an algorithmic tree search mechanism, leveraging an agent skill framework to enable the collaborative management of diverse research concepts and structured experimental exploration while continuously refining research directions and preserving valid hypotheses. Evaluated on the MIPLib and MINLPLib benchmarks, AutoMIP discovers numerous new optimal solutions and achieves success rates that significantly surpass those of existing automated research baselines.

This work addresses the challenge of solving mixed binary quadratic programming (MBQP) problems, which are notoriously difficult and for which existing heuristics often fail to produce high-quality feasible solutions within limited time. The authors propose a machine learning–based primal heuristic featuring a novel neural network architecture tailored for MBQP, coupled with an efficient training data collection pipeline. To enhance model generalization, they integrate contrastive loss with weighted cross-entropy loss and introduce a cross-regime transfer inference mechanism. Extensive evaluations on both standard and real-world MBQP benchmarks demonstrate that the proposed method significantly outperforms state-of-the-art heuristics and commercial solvers. Furthermore, it exhibits strong generalization capabilities in practical applications, notably in wind farm layout optimization.

本文提出一种新方法，通过预测变量在早期搜索阶段和最终解之间的一致性来加速MILP求解过程，从而提高求解效率和质量。

This work proposes DeepBound, a deep learning–based node selection algorithm for mixed-integer linear programming (MILP) that overcomes the limited generalization and instability of traditional handcrafted heuristics. By leveraging a multi-level feature fusion network and a pairwise training paradigm, DeepBound automatically learns optimal branching strategies from data, effectively mitigating node imbalance in branch-and-bound trees. Experimental results on three NP-hard MILP benchmarks demonstrate that DeepBound significantly outperforms both classical heuristics and existing learning-based methods, achieving substantially faster solution times, quicker discovery of high-quality feasible solutions, and strong generalization performance on large-scale, complex instances.

This study investigates whether solutions obtained by reformulating bilevel linear programs into single-level mixed-integer linear programs (MILPs) via the KKT conditions and the Big-M method retain bilevel optimality. We establish, for the first time, that verifying the bilevel optimality of an MILP solution is coNP-complete if even a single Big-M parameter is improperly chosen. Moreover, we show that confirming the global correctness of all Big-M values remains computationally intractable, even when an optimal MILP solution is given. Through complexity-theoretic analysis, we derive two complementary computational lower bounds, demonstrating the inherent intractability of this verification problem. These results apply broadly to uncoupled min-max problems and integer bilevel programs reformulated using strong duality.