Applying a Random-Key Optimizer on Mixed Integer Programs

Configuring the hyperparameters of Mixed-integer programming (MIP) solvers is a high-dimensional, instance-dependent optimization problem where suboptimal settings can degrade solving time by orders of magnitude. Default configurations are often suboptimal, while traditional tuning methods either suffer from the ``cold-start'' problem and inefficient search or heavily rely on expert experience. This paper introduces GRIMIP (General Reasoning for Instance-specific MIP configuration), a novel hybrid intelligence framework that synergistically integrates the semantic reasoning capabilities of Large Language Models (LLMs) with the sample-efficient search of Bayesian Optimization (BO). GRIMIP enables the LLM to function as a complete probabilistic surrogate within the BO loop, significantly improving performance and reducing sampling and evaluation costs. On seven benchmarks including MIPLIB, GRIMIP achieves over 40% reduction in Primal-Dual Integral on hard instances, outperforming SMAC and other LLM-assisted BO methods. By granting LLMs sufficient autonomy, GRIMIP combines the expert-level reasoning of LLMs with the efficient search of BO, achieving state-of-the-art performance.

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. problem-agnostic and can accommodate a broad class of mixed-integer optimization problems through suitable projection operators. 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.

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.