A Preconditioned Iterative Interior Point Approach to the Conic Bundle Subproblem

TL;DR

This paper introduces a preconditioned iterative interior point method for solving large-scale semidefinite programming subproblems more efficiently by exploiting low-rank structures and proposing two novel preconditioning strategies.

Contribution

It develops and analyzes two new preconditioning approaches for iterative solvers in interior point methods, improving scalability for large semidefinite programming problems.

Findings

Deterministic preconditioner effectively controls condition number.

Iterative solver outperforms direct methods for large instances with moderate precision.

Structural low-rank properties can be exploited to enhance solver efficiency.

Abstract

The conic bundle implementation of the spectral bundle method for large scale semidefinite programming solves in each iteration a semidefinite quadratic subproblem by an interior point approach. For larger cutting model sizes the limiting operation is collecting and factorizing a Schur complement of the primal-dual KKT system. We explore possibilities to improve on this by an iterative approach that exploits structural low rank properties. Two preconditioning approaches are proposed and analyzed. Both might be of interest for rank structured positive definite systems in general. The first employs projections onto random subspaces, the second projects onto a subspace that is chosen deterministically based on structural interior point properties. For both approaches theoretic bounds are derived for the associated condition number. In the instances tested the deterministic preconditioner provides surprisingly efficient control on the actual condition number. The results suggest that for large scale instances the iterative solver is usually the better choice if precision requirements are moderate or if the size of the Schur complemented system clearly exceeds the active dimension within the subspace giving rise to the cutting model of the bundle method.