Low-precision first-order method-based fix-and-propagate heuristics for large-scale mixed-integer linear optimization

TL;DR

The paper shows that low-precision first-order methods can be effectively embedded into fix-and-propagate heuristics to solve large-scale mixed-integer linear programs. By solving LP relaxations with a PDHG-based solver (PDLP) to low accuracy and integrating the resulting information into multiple FP variants, the authors obtain high-quality feasible solutions for large MIPs, including energy-system unit-commitment models. Experiments on MIPLIB 2017 and REMix-based ESOMs demonstrate that low-accuracy LP solutions do not degrade FP performance; PDLP-based FP achieves gaps often below 2% in under 4 hours on the largest instances, whereas commercial solvers struggle or fail. The work provides a foundation for GPU-accelerated MIP solving using first-order methods and opens avenues for further FOM-based heuristics and GPU-enabled domain propagation and search strategies.

Abstract

We investigate the use of low-precision first-order methods (FOMs) within a fix-and-propagate (FP) framework for solving mixed-integer programming problems (MIPs). FOMs, using only matrix-vector products instead of matrix factorizations, are well suited for GPU acceleration and have recently gained more attention for their application to large-scale linear programming problems (LPs). We employ PDLP, a variant of the Primal-Dual Hybrid Gradient (PDHG) method specialized to LP problems, to solve the LP-relaxation of our MIPs to low accuracy. This solution is used to motivate fixings within our fix-and-propagate framework. We implemented four different FP variants using primal and dual LP solution information. We evaluate the performance of our heuristics on MIPLIB 2017, showcasing that the low-accuracy LP solution produced by the FOM does not lead to a loss in quality of the FP heuristic solutions when compared to a high-accuracy interior-point method LP solution. Further, we use our FP framework to produce high-accuracy solutions for large-scale (up to 243 million non-zeros and 8 million decision variables) unit-commitment energy-system optimization models created with the modeling framework REMix. For the largest problems, we can generate solutions with under 2% primal-dual gap in less than 4 hours, whereas commercial solvers cannot generate feasible solutions within two days of runtime. This study represents the first successful application of FOMs in large-scale mixed-integer optimization, demonstrating their efficacy and establishing a foundation for future research in this domain.