Optimization over Structured Subsets of Positive Semidefinite Matrices via Column Generation
This work provides a new algorithmic framework for approximating semidefinite cones, which is relevant for optimization problems in control, combinatorial optimization, and polynomial optimization.
The authors develop iterative algorithms to inner-approximate the positive semidefinite cone using linear and second-order cone programming, improving upon an initial approximation via column generation. They apply these techniques to approximate the sum of squares cone for polynomial optimization and the copositive cone for discrete optimization.
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.