A Hybrid Direct-Iterative Method for Solving KKT Linear Systems

TL;DR

This paper introduces a hybrid method combining iterative and direct solves to efficiently handle KKT systems in interior point methods, optimizing GPU performance by reducing communication costs compared to traditional LDL^T factorization.

Contribution

The paper presents a novel hybrid approach that solves large indefinite systems by decomposing them into smaller positive definite systems, enabling efficient GPU implementation without pivoting.

Findings

Outperforms LDL^T factorization on large systems

Reduces communication costs on GPUs

Enables reuse of symbolic factorization

Abstract

We propose a solution strategy for linear systems arising in interior method optimization, which is suitable for implementation on hardware accelerators such as graphical processing units (GPUs). The current gold standard for solving these systems is the LDL^T factorization. However, LDL^T requires pivoting during factorization, which substantially increases communication cost and degrades performance on GPUs. Our novel approach solves a large indefinite system by solving multiple smaller positive definite systems, using an iterative solve for the Schur complement and an inner direct solve (via Cholesky factorization) within each iteration. Cholesky is stable without pivoting, thereby reducing communication and allowing reuse of the symbolic factorization. We demonstrate the practicality of our approach and show that on large systems it can efficiently utilize GPUs and outperform LDL^T factorization of the full system.