Inner approximating the completely positive cone via the cone of scaled diagonally dominant matrices

TL;DR

This paper introduces a new inner approximation for the completely positive cone using scaled diagonally dominant matrices, enabling efficient graph-based second-order cone schemes that balance expressiveness and computational speed.

Contribution

It proposes the cone of nonnegative scaled diagonally dominant matrices as a novel inner approximation for the completely positive cone, with new hierarchy schemes for optimization.

Findings

Effective approximation of the completely positive cone.

Balanced trade-off between SDP expressiveness and LP speed.

Numerical results show improved performance on standard problems.

Abstract

Motivated by the expressive power of completely positive programming to encode hard optimization problems, many approximation schemes for the completely positive cone have been proposed and successfully used. Most schemes are based on outer approximations, with the only inner approximations available being linear programming based methods proposed by Bundfuss and D\"ur and also Y{\i}ld{\i}r{\i}m, and a semidefinite programming based method proposed by Lasserre. In this paper, we propose the use of the cone of nonnegative scaled diagonally dominant matrices as a natural inner approximation to the completely positive cone. Using projections of this cone we derive new graph-based second-order cone approximation schemes for completely positive programming, leading to both uniform and problem-dependent hierarchies. This offers a compromise between the expressive power of semidefinite programming and the speed of linear programming based approaches. Numerical results on random problems, standard quadratic programs and the stable set problem are presented to illustrate the effectiveness of our approach.