SDSL-Solver: Scalable Distributed Sparse Linear Solvers for Large-Scale Interior Point Methods

TL;DR

SDSL-Solver is a scalable distributed framework for solving large sparse linear systems in interior point methods, significantly reducing computation time on multi-node clusters.

Contribution

It introduces a novel distributed solver with two parallel methods and preconditioner reuse, improving scalability and efficiency for large-scale optimization problems.

Findings

Achieves up to 97.54x speedup over single-node PARDISO.

Attains 6.23x and 7.77x speedups over PETSc on four nodes.

Effectively handles matrices with millions of variables.

Abstract

The solution of sparse linear systems constitutes the dominant computational bottleneck in interior point methods (IPMs), frequently consuming over 70% of the total solution time. As optimization problems scale to millions of variables, direct solvers encounter prohibitive fill-in, excessive memory consumption, and limited parallel scalability. We present SDSL-Solver, a scalable distributed sparse linear solver framework designed for IPMs. SDSL-Solver employs Krylov subspace methods, combined with numerics-based sparse filtering and diagonal correction techniques that produce high-quality preconditioners. To accommodate diverse problem characteristics, SDSL-Solver offers two complementary distributed parallel methods: Block Jacobi for diagonally dominant matrices, and Bordered Block Diagonal (BBD) for general or ill-conditioned matrices requiring globally coupled preconditioning via Schur complement techniques. A preconditioner reuse strategy further amortizes construction costs across consecutive IPMs iterations. We evaluate SDSL-Solver on benchmark problems with matrix dimensions ranging from tens of thousands to over five million on multi-node clusters equipped with X86 processors. The experimental results show that under the Block Jacobi and BBD distributed methods, SDSL-Solver on a four-node configuration achieves average speedups of 6.23 times and 7.77 times, respectively, compared to PETSc running on the same number of nodes. Relative to the single-node PARDISO, the average speedups reach 97.54 times and 5.85 times, respectively.