Barrier and penalty methods for low-rank semidefinite programming with application to truss topology design

TL;DR

This paper introduces a novel preconditioned conjugate gradient approach with a specialized preconditioner for efficiently solving large, sparse low-rank semidefinite programs, demonstrated through truss topology optimization.

Contribution

It presents a new preconditioning technique that leverages low-rank structure, applicable within interior-point and primal-dual penalty methods for SDPs.

Findings

Preconditioner significantly accelerates convergence in large SDPs.

Method effectively handles high-dimensional truss topology problems.

Numerical results show improved computational efficiency.

Abstract

The aim of this paper is to solve large-and-sparse linear Semidefinite Programs (SDPs) with low-rank solutions. We propose to use a preconditioned conjugate gradient method within second-order SDP algorithms and introduce a new efficient preconditioner fully utilizing the low-rank information. We demonstrate that the preconditioner is universal, in the sense that it can be efficiently used within a standard interior-point algorithm, as well as a newly developed primal-dual penalty method. The efficiency is demonstrated by numerical experiments using the truss topology optimization problems of growing dimension.