Accelerating MPGP-type Methods Through Preconditioning

TL;DR

This paper explores a novel approximate preconditioning method to accelerate MPGP-type algorithms for quadratic programming, achieving significant speedups while maintaining theoretical bounds on error and condition number.

Contribution

It introduces an approximate variant of preconditioning in face that reduces computational cost by updating the inner preconditioner only once, with theoretical analysis and numerical validation.

Findings

Large speedups demonstrated in numerical experiments.

Sharp bounds on condition number of the preconditioned operator.

Approximate preconditioning maintains accuracy with reduced updates.

Abstract

This work investigates the acceleration of MPGP-type algorithms using preconditioning for the solution of quadratic programming problems. The preconditioning needs to be done only on the free set so as not to change the constraints. A variant of preconditioning restricted to the free set is the preconditioning in face. The inner preconditioner in preconditioning in face needs to be recomputed or updated every time the free set changes. Here, we investigate an approximate variant of preconditioning in face that computes the inner preconditioner only once. We analyze the error of the approximate variant, give a sharp bound on the condition number of the preconditioned operator, and provide numerical experiments demonstrating that very large speedups can be achieved by the approximate variant.