Decomposition of arrow type positive semidefinite matrices with application to topology optimization

TL;DR

This paper introduces a method to decompose arrow-type positive semidefinite matrices, reducing problem size and computational complexity in large-scale semidefinite optimization, with applications to topology optimization.

Contribution

It shows that for arrow matrices with certain properties, the introduced dense matrix variables can be replaced by vectors, significantly improving computational efficiency.

Findings

Numerical examples demonstrate linear growth in complexity after decomposition.

Decomposed problems solve faster than original large-scale problems.

The approach connects to domain decomposition and Steklov-Poincaré operators.

Abstract

Decomposition of large matrix inequalities for matrices with chordal sparsity graph has been recently used by Kojima et al.\ \cite{kim2011exploiting} to reduce problem size of large scale semidefinite optimization (SDO) problems and thus increase efficiency of standard SDO software. A by-product of such a decomposition is the introduction of new dense small-size matrix variables. We will show that for arrow type matrices satisfying suitable assumptions, the additional matrix variables have rank one and can thus be replaced by vector variables of the same dimensions. This leads to significant improvement in efficiency of standard SDO software. We will apply this idea to the problem of topology optimization formulated as a large scale linear semidefinite optimization problem. Numerical examples will demonstrate tremendous speed-up in the solution of the decomposed problems, as compared to the original large scale problem. In our numerical example the decomposed problems exhibit linear growth in complexity, compared to the more than cubic growth in the original problem formulation. We will also give a connection of our approach to the standard theory of domain decomposition and show that the additional vector variables are outcomes of the corresponding discrete Steklov-Poincar\'{e} operators.