Decomposition in conic optimization with partially separable structure

TL;DR

This paper introduces a decomposition method for conic optimization problems with partially separable structures, enabling more efficient solutions for applications like sparse semidefinite programming with chordal sparsity.

Contribution

It extends decomposition techniques to non-polyhedral conic problems by leveraging partial separability and combines Spingarn's method with interior-point techniques.

Findings

Effective decomposition for conic problems with partial separability

Application to sparse semidefinite programming with chordal sparsity

Improved computational efficiency in solving large-scale conic problems

Abstract

Decomposition techniques for linear programming are difficult to extend to conic optimization problems with general non-polyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables. However in many applications the convex cones have a partially separable structure that allows them to be characterized in terms of simpler lower-dimensional cones. The most important example is sparse semidefinite programming with a chordal sparsity pattern. Here partial separability derives from the clique decomposition theorems that characterize positive semidefinite and positive-semidefinite-completable matrices with chordal sparsity patterns. The paper describes a decomposition method that exploits partial separability in conic linear optimization. The method is based on Spingarn's method for equality constrained convex optimization, combined with a fast interior-point method for evaluating proximal operators.