Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Marilena Palomba, Lucas Slot, Luis Felipe Vargas, Monaldo Mastrolilli

TL;DR

This paper analyzes the computational complexity of sum-of-squares (SoS) relaxations for copositive programs and identifies precise conditions under which the resulting semidefinite programs (SDPs) are solvable in polynomial time up to fixed precision. The authors prove that, if a copositive program satisfies a polynomially bounded optimal solution (PBOS) and possesses an interior SPN point (intSPN), then the SOS relaxations $p_K^{(r)}$ and $p_Q^{(r)}$ can be computed in time poly$(n,\log 1/\varepsilon)$ for fixed $r$, and that these conditions hold for standard quadratic programs and their reciprocals. They also show how to modify a copositive program to enforce PBOS via a construction $\mathrm{CP-1}$, enabling polynomial-time computation of the relaxations and providing applications to the weighted stability number and, via a reformulation, to the chromatic number of graphs. The work further establishes bounds on SOS coefficients to keep SDPs of polynomial size and presents pathological examples to illustrate the necessity of the assumptions. Overall, the results provide a theoretical foundation for efficiently computing SOS-based bounds in copositive programming and yield practical polynomial-time bounds for important optimization problems.

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 Peña, 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.

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

TL;DR

This paper analyzes the computational complexity of sum-of-squares (SoS) relaxations for copositive programs and identifies precise conditions under which the resulting semidefinite programs (SDPs) are solvable in polynomial time up to fixed precision. The authors prove that, if a copositive program satisfies a polynomially bounded optimal solution (PBOS) and possesses an interior SPN point (intSPN), then the SOS relaxations and can be computed in time poly for fixed , and that these conditions hold for standard quadratic programs and their reciprocals. They also show how to modify a copositive program to enforce PBOS via a construction , enabling polynomial-time computation of the relaxations and providing applications to the weighted stability number and, via a reformulation, to the chromatic number of graphs. The work further establishes bounds on SOS coefficients to keep SDPs of polynomial size and presents pathological examples to illustrate the necessity of the assumptions. Overall, the results provide a theoretical foundation for efficiently computing SOS-based bounds in copositive programming and yield practical polynomial-time bounds for important optimization problems.

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 Peña, 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.
Paper Structure (24 sections, 17 theorems, 133 equations)

This paper contains 24 sections, 17 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Sum-of-squares relaxations
  3. Our contributions
  4. Technical overview
  5. Related work
  6. Preliminaries
  7. Notations
  8. Bit size
  9. Semidefinite programming
  10. Supremum norm and coefficient norm
  11. Bounding the coefficients in sum-of-squares proofs
  12. Proof of \ref{['THM:main']}
  13. Polynomial-time computation of CP-K
  14. Polynomial-time computation of CP-Q
  15. Application to standard quadratic programs
  16. ...and 9 more sections

Key Result

Theorem 1.3

Let $C, A_1, \ldots, A_m$ be the matrices defining the copositive program cop-program. Suppose that the following assumptions hold: Then, for any fixed $r \in \mathbb N$, the parameter $p_\mathcal{K}^{{(r)}}$ (resp. $p_\mathcal{Q}^{{(r)}}$) can be computed up to an additive error $\varepsilon > 0$ in time $\mathrm{poly}(n, \log(1/\varepsilon$)).

Theorems & Definitions (34)

  • Example 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: dKV16, see also GPS23
  • Lemma 2.2: KORDA20171
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • ...and 24 more