Constrained Graph Diffusion for Mixed-Integer Optimization

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. We introduce Constrained Graph Diffusion (CGD), a learning-based framework that approximately solves recurring instances of such problems by learning a conditional distribution over their discrete decisions. CGD uses a graph-based diffusion model and incorporates constraint information directly into the reverse diffusion process, steering intermediate predictions toward the feasible region throughout generation. By operating on continuous relaxations of the discrete variables, CGD defines a differentiable constrained generation pathway up to terminal discrete recovery. Once the discrete decision is recovered and fixed, a numerical optimizer solves the remaining continuous problem, avoiding online combinatorial search over the binary variables while retaining numerical optimization for continuous completion. We evaluate CGD on AC-OPF with branch switching and discrete portfolio optimization, demonstrating substantial improvements in feasibility and solution quality over learning-based baselines while achieving speedups of up to $543\times$ over state-of-the-art MIP solvers on large instances.