Batched First-Order Methods for Parallel LP Solving in MIP

2026-01-29

AI Summary

This work addresses the challenge of efficiently solving numerous related linear programming (LP) subproblems arising in mixed-integer programming (MIP), particularly in contexts such as strong branching and bound tightening, where traditional methods fail to exploit GPU parallelism effectively. The authors propose a GPU-oriented batched first-order optimization method, reformulating the primal–dual hybrid gradient algorithm into matrix–matrix operations to significantly enhance parallel efficiency. This study represents the first systematic application of batched first-order methods to LP subproblems within MIP solvers, advocating that GPUs should perform core computational tasks rather than merely assist CPU-based heuristics. The approach promotes deeper co-design between MIP algorithms and GPU architectures. Experimental results demonstrate that the proposed method outperforms conventional simplex solvers under specific problem scales and hardware configurations.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We present a batched first-order method for solving multiple linear programs in parallel on GPUs. Our approach extends the primal-dual hybrid gradient algorithm to efficiently solve batches of related linear programming problems that arise in mixed-integer programming techniques such as strong branching and bound tightening. By leveraging matrix-matrix operations instead of repeated matrix-vector operations, we obtain significant computational advantages on GPU architectures. We demonstrate the effectiveness of our approach on various case studies and identify the problem sizes where first-order methods outperform traditional simplex-based solvers depending on the computational environment one can use. This is a significant step for the design and development of integer programming algorithms tightly exploiting GPU capabilities where we argue that some specific operations should be allocated to GPUs and performed in full instead of using light-weight heuristic approaches on CPUs.

Problem

• parallel LP solving
• mixed-integer programming
• GPU acceleration
• batched linear programs
• strong branching

Innovation

• batched first-order methods
• GPU acceleration
• primal-dual hybrid gradient
• parallel LP solving
• mixed-integer programming