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

@article{Laurent2021ExactnessOP, 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={2021} }

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…

## 5 Citations

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

- MathematicsSIAM 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.…

### On the Exactness of Sum-of-Squares Approximations for the Cone of $5\times 5$ Copositive Matrices

- Mathematics
- 2022

We investigate the hierarchy of conic inner approximations K ( r ) n ( r ∈ N ) for the copositive cone COP n , introduced by Parrilo ( Structured Semideﬁnite Programs and Semialgebraic Geometry…

### On the exactness of sum-of-squares approximations for the cone of 5 × 5 copositive matrices

- MathematicsLinear Algebra and its Applications
- 2022

### On the structure of the $6 \times 6$ copositive cone

- Mathematics
- 2022

In this work we complement the description of the extreme rays of the 6 × 6 copositive cone with some topological structure. In a previous paper we decomposed the set of extreme elements of this cone…

### A Sum of Squares Characterization of Perfect Graphs

- Mathematics
- 2021

We present an algebraic characterization of perfect graphs, i.e., graphs for which the clique number and the chromatic number coincide for every induced subgraph. We show that a graph is perfect if…

## References

SHOWING 1-10 OF 32 REFERENCES

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

- Mathematics, Computer ScienceSIAM 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

- MathematicsMath. 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

- MathematicsSIAM 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.…

### On the exactness of sum-of-squares approximations for the cone of 5 × 5 copositive matrices

- MathematicsLinear Algebra and its Applications
- 2022

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

- Computer Science, MathematicsSIAM 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

- MathematicsMath. Program.
- 1997

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 ScienceMath. 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.

### Generating irreducible copositive matrices using the stable set problem

- MathematicsDiscret. Appl. Math.
- 2021

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

- MathematicsMath. Oper. Res.
- 2003

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.

### On standard quadratic programs with exact and inexact doubly nonnegative relaxations

- Computer Science, MathematicsMath. Program.
- 2022

This work presents a full algebraic characterization of the set of instances of standard quadratic programs that admit an exact doubly nonnegative relaxation, and establishes several relations between the so-called convexity graph of an instance and the tightness of the doublynonnegative relaxation.