Efficient Preconditioners for Interior Point Methods via a new Schur Complement-Based Strategy

TL;DR

This paper introduces a new Schur complement-based preconditioning strategy for interior point methods that improves convergence and computational efficiency in solving constrained convex quadratic programming problems.

Contribution

The paper presents a novel preconditioned inexact primal-dual interior point method using a reduced Schur complement system and two new preconditioners that enhance solver performance.

Findings

Preconditioned methods reduce eigenvalues and improve convergence.

Significant cost reduction compared to existing methods.

Effective for problems with small numbers of constraints or degrees of freedom.

Abstract

We propose a novel preconditioned inexact primal-dual interior point method for constrained convex quadratic programming problems. The algorithm we describe invokes the preconditioned conjugate gradient method on a new reduced Schur complement KKT system, in implicit form. In contrast to standard approaches, the Schur complement formulation we consider enables reuse of the factorization of the KKT matrix with rows and columns corresponding to inequality constraints excluded, across all interior point iterations. Further, two new preconditioners are presented for the resulting reduced system, that alleviate the ill-conditioning associated with slack variables in primal-dual interior point methods. Each of the preconditioners we propose also provably reduces the number of unique eigenvalues for the coefficient matrix, and thus the CG iteration count. One preconditioner is efficient when the number of equality constraints is small, while the other is efficient when the number of remaining degrees of freedom is small. Numerical experiments with synthetic problems and problems from the Maros-M\'esz\'aros QP collection show that our preconditioned inexact interior point solvers are effective at improving conditioning and reducing cost. Across all test problems for which the direct method is not fastest, our preconditioned methods achieve a reduction in cost by a geometric mean of $1.432$ relative to the best alternative preconditioned method for each problem.