Batched First-Order Methods for Parallel LP Solving in MIP

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. This is a significant step toward integer programming algorithms that tightly exploit GPU capabilities. We argue that some specific operations should be allocated to GPUs and performed in full, instead of relying on lightweight heuristic approaches on CPUs.