On Polyhedral and Second-Order Cone Decompositions of Semidefinite Optimization Problems

TL;DR

This paper analyzes a cutting-plane method for semidefinite optimization, proving its convergence and demonstrating its effectiveness with second-order cone initializations on large-scale problems.

Contribution

It provides a convergence proof for the cutting-plane method in SDOs and shows improved performance using second-order cone approximations.

Findings

Achieved bound gaps of 0.5-6.5% in sparse PCA problems with thousands of covariates.

Successfully solved nuclear norm problems over 500x500 matrices.

Demonstrated the advantage of second-order cone initialization over linear approximation.

Abstract

We study a cutting-plane method for semidefinite optimization problems (SDOs), and supply a proof of the method's convergence, under a boundedness assumption. By relating the method's rate of convergence to an initial outer approximation's diameter, we argue that the method performs well when initialized with a second-order-cone approximation, instead of a linear approximation. We invoke the method to provide bound gaps of 0.5-6.5% for sparse PCA problems with $1000$s of covariates, and solve nuclear norm problems over 500x500 matrices.