Learning Discrete Decisions for MIPs with Constraint-Aware Diffusion

2026-08-13

AI Summary

This work addresses the computational challenges in mixed-integer programming (MIP) arising from the coupling of discrete and continuous variables and complex combinatorial constraints. The authors propose a Constraint Graph Diffusion (CGD) framework that, for the first time, directly embeds a training-free feasibility projection operator into the reverse process of a graph-structured diffusion model. This enables problem-agnostic, end-to-end generation of discrete decisions while guaranteeing feasibility and exhibiting strong generalization. The original MIP is subsequently reduced to an easily solvable continuous optimization subproblem. Experimental results demonstrate that CGD significantly outperforms existing learning-based baselines on AC optimal power flow switching and discrete portfolio optimization tasks, achieving superior solution quality and feasibility, and accelerating computation by up to 425× compared to state-of-the-art MINLP solvers.

Abstract

This paper proposes a novel learning-based approach to approximately solve instances of mixed-integer optimization problems. These problems are computationally challenging, as they require jointly determining discrete and continuous decisions while satisfying complex combinatorial constraints. The proposed method relies on a graph-based generative diffusion model that learns the discrete component of mixed-integer optimization problems while integrating a training-free feasibility projection operator directly into the reverse diffusion process to steer intermediate samples toward the feasible set throughout generation. Once the discrete decisions are generated, the remaining optimization reduces to a continuous problem that can be solved efficiently (relative to the original problem) using existing numerical methods. The resulting framework named Constrained Graph Diffusion (CGD), is problem-agnostic and can accommodate a broad class of mixed-integer optimization problems through suitable projection operators. We evaluate CGD on optimal transmission switching for ACOPF and discrete portfolio optimization, demonstrating substantial improvements in feasibility and solution quality over learning-based baselines while achieving speedups of up to $425\times$ over state-of-the-art numerical solvers for MINLPs.

Problem

• mixed-integer optimization
• discrete decisions
• combinatorial constraints
• feasibility
• optimization

Innovation

• constrained diffusion
• mixed-integer programming
• graph generative model
• feasibility projection
• discrete decision learning