Optimization over Structured Subsets of Positive Semidefinite Matrices via Column Generation

TL;DR

This paper introduces iterative algorithms using column generation to improve inner approximations of positive semidefinite cones, with applications to polynomial and discrete optimization.

Contribution

It presents novel algorithms that enhance inner approximations of positive semidefinite cones via column generation, applicable to sum of squares and copositive cones.

Findings

Algorithms improve approximation accuracy at each iteration

Applications to polynomial optimization problems

Effective approximation of copositive cones

Abstract

We develop algorithms for inner approximating the cone of positive semidefinite matrices via linear programming and second order cone programming. Starting with an initial linear algebraic approximation suggested recently by Ahmadi and Majumdar, we describe an iterative process through which our approximation is improved at every step. This is done using ideas from column generation in large-scale linear and integer programming. We then apply these techniques to approximate the sum of squares cone in a nonconvex polynomial optimization setting, and the copositive cone for a discrete optimization problem.