A New Preconditioning Approach for an Interior Point-Proximal Method of Multipliers for Linear and Convex Quadratic Programming

TL;DR

This paper introduces a novel preconditioning strategy for an interior point-proximal method of multipliers, improving the efficiency and robustness of solving large-scale linear and quadratic programming problems.

Contribution

It proposes a new preconditioning approach based on sparsification and block-diagonal structures, enhancing solver performance for large-scale problems.

Findings

Effective preconditioning accelerates convergence of iterative solvers.

Robustness demonstrated on large-scale test problems.

Method reduces computational resources needed for large systems.

Abstract

In this paper, we address the efficient numerical solution of linear and quadratic programming problems, often of large scale. With this aim, we devise an infeasible interior point method, blended with the proximal method of multipliers, which in turn results in a primal-dual regularized interior point method. Application of this method gives rise to a sequence of increasingly ill-conditioned linear systems which cannot always be solved by factorization methods, due to memory and CPU time restrictions. We propose a novel preconditioning strategy which is based on a suitable sparsification of the normal equations matrix in the linear case, and also constitutes the foundation of a block-diagonal preconditioner to accelerate MINRES for linear systems arising from the solution of general quadratic programming problems. Numerical results for a range of test problems demonstrate the robustness of the proposed preconditioning strategy, together with its ability to solve linear systems of very large dimension.