Solving Integer Linear Programming with Parallel Tempering

Efficient branching policies are essential for accelerating Mixed Integer Linear Programming (MILP) solvers. Their design has long relied on hand-crafted heuristics, and now machine learning has emerged as a promising paradigm to automate this process. However, existing learning-based methods are often hindered by their dependence on expensive expert demonstrations and the gap between training objectives and the solver's end-to-end performance. In this work, we propose LLM4Branch, a novel framework that leverages Large Language Models (LLMs) to automate the discovery of efficient branching policies. Specifically, the discovered policy is an executable program with a program skeleton generated by the LLM and a parameter vector, which is optimized via a zeroth-order method over a few instances with their end-to-end performance feedback. Extensive experiments on standard MILP benchmarks demonstrate that LLM4Branch establishes a new state-of-the-art among CPU-based methods and achieves performance competitive with advanced GPU-based models. Codes are available at https://github.com/hzn18/LLM4Branch.

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 \textbf{GRIMIP} (\textbf{\underline{G}}eneral \textbf{\underline{R}}easoning for \textbf{\underline{I}}nstance-specific \textbf{\underline{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.

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.