Computational complexity of sum-of-squares bounds for copositive programs

TL;DR

This paper investigates the computational complexity of sum-of-squares relaxations for copositive programs, establishing conditions for polynomial-time solvability and illustrating both practical applications and theoretical limitations.

Contribution

It provides sufficient conditions for polynomial-time computation of SOS relaxations of copositive programs and analyzes their complexity in standard and pathological cases.

Findings

SOS relaxations are polynomial-time computable under certain conditions.

Standard quadratic programs satisfy the conditions for efficient SOS bounds.

Pathological examples show SOS relaxations can require doubly-exponential size solutions.

Abstract

In recent years, copositive programming has received significant attention for its ability to model hard problems in both discrete and continuous optimization. Several relaxations of copositive programs based on semidefinite programming (SDP) have been proposed in the literature, meant to provide tractable bounds. However, while these SDP-based relaxations are amenable to the ellipsoid algorithm and interior point methods, it is not immediately obvious that they can be solved in polynomial time (even approximately). In this paper, we consider the sum-of-squares (SOS) hierarchies of relaxations for copositive programs introduced by Parrilo (2000), de Klerk & Pasechnik (2002) and Pe\~na, Vera & Zuluaga (2006), which can be formulated as SDPs. We establish sufficient conditions that guarantee the polynomial-time computability (up to fixed precision) of these relaxations. These conditions are satisfied by copositive programs that represent standard quadratic programs and their reciprocals. As an application, we show that the SOS bounds for the (weighted) stability number of a graph can be computed efficiently. Additionally, we provide pathological examples of copositive programs (that do not satisfy the sufficient conditions) whose SOS relaxations admit only feasible solutions of doubly-exponential size.