# Exactness of Parrilo’s Conic Approximations for Copositive Matrices and Associated Low Order Bounds for the Stability Number of a Graph

@article{Laurent2022ExactnessOP,
title={Exactness of Parrilo’s Conic Approximations for Copositive Matrices and Associated Low Order Bounds for the Stability Number of a Graph},
author={Monique Laurent and Luis Felipe Vargas},
journal={Mathematics of Operations Research},
year={2022}
}
• Published 27 September 2021
• Mathematics
• Mathematics of Operations Research
De Klerk and Pasechnik introduced in 2002 semidefinite bounds [Formula: see text] for the stability number [Formula: see text] of a graph G and conjectured their exactness at order [Formula: see text]. These bounds rely on the conic approximations [Formula: see text] introduced by Parrilo in 2000 for the copositive cone [Formula: see text]. A difficulty in the convergence analysis of the bounds is the bad behavior of Parrilo’s cones under adding a zero row/column: when applied to a matrix not…

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## References

SHOWING 1-10 OF 32 REFERENCES

### Approximation of the Stability Number of a Graph via Copositive Programming

• Mathematics, Computer Science
SIAM J. Optim.
• 2002
This paper shows how the stability number can be computed as the solution of a conic linear program (LP) over the cone of copositive matrices of a graph by solving semidefinite programs (SDPs) of increasing size (lift-and-project method).

### Semidefinite bounds for the stability number of a graph via sums of squares of polynomials

• Mathematics
Math. Program.
• 2005
The hierarchy of Lasserre is known to converge in α(G) steps as it refines the hierarchy of Lovasz and Schrijver, and the de Klerk and Pasechnik conjecture that their hierarchy also finds the stability number after α( G) steps is proved.

### Finite Convergence of Sum-of-Squares Hierarchies for the Stability Number of a Graph

• Mathematics
SIAM J. Optim.
• 2022
We investigate a hierarchy of semidefinite bounds θ(G) for the stability number α(G) of a graph G, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [SIAM J.

### Computing the Stability Number of a Graph Via Linear and Semidefinite Programming

• Computer Science, Mathematics
SIAM J. Optim.
• 2007
This work is based on and refines de Klerk and Pasechnik’s approach to approximating the stability number via copositive programming and provides a closed-form expression for the values computed by the linear programming approximations.

### An exact duality theory for semidefinite programming and its complexity implications

In this paper, an exact dual is derived for Semidefinite Programming (SDP), for which strong duality properties hold without any regularity assumptions, and the dual is then applied to derive certain complexity results for SDP.

### Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs

• Mathematics, Computer Science
Math. Program.
• 2020
This work studies a class of quadratically constrained quadratic programs (QCQPs), called diagonal QCQPs, and provides a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact, the first result establishing the exactness of the semideFinite relaxation for random general QCZPs.

### Continuous Characterizations of the Maximum Clique Problem

• Mathematics
Math. Oper. Res.
• 1997
The Motzkin-Strauss QP is characterized, revealing interesting underlying discrete structures, and its properties are investigated, which are polynomial time verifiable.

### A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0-1 Programming

The three methods for constructing hierarchies of successive linear or semidefinite relaxations of a polytope are presented in a common elementary framework and it is shown that the Lasserre construction provides the tightest Relaxations of P.