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

2025-01-07

AI Summary

This paper investigates the computational complexity of sum-of-squares (SOS) relaxations in copositive programming. Focusing on standard quadratic programming and its reciprocal problem, it establishes the **first exact sufficient condition** under which SOS relaxations are solvable in polynomial time, and characterizes the boundary of infeasibility; when this condition fails, pathological instances arise where solution size exhibits double-exponential blowup. Methodologically, the work integrates the SOS hierarchy, semidefinite programming (SDP), and the ellipsoid method, and introduces a weighted modeling framework for the stability number. Key results show that the SOS bound is polynomial-time approximable, and the (weighted) stability number of a graph admits an efficiently computable SOS upper bound. This work provides the first systematic characterization of the tractability boundary and intrinsic computational nature of SOS methods in copositive optimization.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Computational social science
• Security and Privacy: Large-scale security measurements

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.

Problem

• Sum of Squares (SOS) methods
• Computational complexity
• Positive semidefinite programming

Innovation

• SOS Relaxation
• Quadratic Programming
• Graph Stability Number