milp formulation design

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 information bottleneck arising from inadequate joint modeling of integer and continuous variables in mixed-integer linear programming (MILP), this paper proposes FMIP—the first generative framework that jointly models their joint distribution. FMIP leverages invertible normalizing flows for end-to-end probabilistic modeling of both variable types and introduces a gradient-driven holistic guidance mechanism that jointly optimizes solution quality and feasibility during inference, while tightly integrating with downstream solvers. Unlike existing ELA methods—which model only integer variables—FMIP breaks from the conventional paradigm, substantially mitigating distribution mismatch. Evaluated on eight standard MILP benchmarks, FMIP reduces the average primal gap by 41.34%. It is compatible with diverse neural architectures and mainstream solvers, achieving a favorable trade-off between computational efficiency and solution optimality.

This paper addresses the compactness of mixed-integer linear programming (MILP) formulations for continuous piecewise-linear (CPWL) functions in arbitrary dimensions. We introduce the notion of “well-behaved” piecewise-linear interpolation and prove that any CPWL function admits an equivalent well-behaved representation—establishing a theoretical foundation for compact modeling. Methodologically, we integrate six compactification strategies: difference-of-convex (DC) representation, variable fixing, auxiliary logical constraints, tightened big-M coefficients, tighter variable bounds, and structural analysis. Experiments demonstrate that their synergistic application substantially improves solver efficiency; in particular, MILP models exploiting the well-behaved structure reduce average solution time by 40%–70% across multiple benchmark instances. Our work provides a systematic, scalable framework for MILP-based optimization of CPWL functions.

Achieving dynamic line balancing in garment manufacturing under multiple operational constraints remains challenging. Method: This paper proposes a synergistic optimization approach integrating Mixed-Integer Linear Programming (MILP) with Lean management principles—marking the first deep coupling of these paradigms for integrated line configuration and order allocation. The model jointly optimizes online/offline order routing, workstation load balancing, and resource constraints, and is deployed at factory scale using IBM CPLEX. Contribution/Results: Empirical validation demonstrates over 50% reduction in labor costs, significant improvements in equipment and workforce utilization, and enhanced order fulfillment capacity. The study not only confirms the scalability and industrial applicability of MILP in garment manufacturing but also establishes a novel line-balancing paradigm that unifies rigorous mathematical modeling with Lean practice—bridging theoretical optimization and operational excellence.

Configuring Mixed-Integer Linear Programming (MILP) solver parameters—particularly for cutting-plane separation—is challenging due to high-dimensional, problem-dependent search spaces; existing machine learning approaches suffer from poor generalization, heavy reliance on large-scale labeled data, and difficulty integrating into solver pipelines. Method: We propose the first LLM-driven zero-shot cutting-plane separator configuration framework. It leverages large language models to jointly parse natural-language problem descriptions and LaTeX-based MILP formulations, augmented by literature-informed prompt engineering and semantic modeling of separators—requiring no custom interfaces or extensive retraining. A lightweight, performance-driven clustering ensemble strategy ensures both robustness and real-time responsiveness. Results: On benchmark combinatorial optimization instances and real-world datasets, our method matches state-of-the-art performance while reducing training data requirements by over 90% and generating configurations in under one second.

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.

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.

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.