Partial facial reduction: simplified, equivalent SDPs via approximations of the PSD cone

TL;DR

This paper introduces a practical facial reduction method for SDPs that uses efficient PSD cone approximations, simplifying problems without strict feasibility and aiding SDP solver performance.

Contribution

It presents a novel, computationally efficient facial reduction technique using PSD cone approximations, with software implementation and methods for rank maximization and sparsity preservation.

Findings

Effective on practical SDPs with no strict feasibility

Provides software for the proposed facial reduction method

Enables maximum rank matrix identification in approximations

Abstract

We develop a practical semidefinite programming (SDP) facial reduction procedure that utilizes computationally efficient approximations of the positive semidefinite cone. The proposed method simplifies SDPs with no strictly feasible solution (a frequent output of parsers) by solving a sequence of easier optimization problems and could be a useful pre-processing technique for SDP solvers. We demonstrate effectiveness of the method on SDPs arising in practice, and describe our publicly-available software implementation. We also show how to find maximum rank matrices in our PSD cone approximations (which helps us find maximal simplifications), and we give a post-processing procedure for dual solution recovery that generally applies to facial-reduction-based pre-processing techniques. Finally, we show how approximations can be chosen to preserve problem sparsity.