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

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 a linear programming based method proposed by Bundfuss and Dur (SIAM J Optim 20(1):30–53, 2009) and also Yildirim (Optim Methods Softw 27(1):155–173, 2012), and a semidefinite programming based method proposed by Lasserre (Math Program 144(1):265–276, 2014). 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.