Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming
Journal of Global Optimization
D.C. Versus Copositive Bounds for Standard QP
Journal of Global Optimization
A quantitative Pólya's Theorem with corner zeros
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
A linear programming reformulation of the standard quadratic optimization problem
Journal of Global Optimization
Effective Pólya semi-positivity for non-negative polynomials on the simplex
Journal of Complexity
Simple ingredients leading to very efficient heuristics for the maximum clique problem
Journal of Heuristics
A quantitative Pólya's Theorem with zeros
Journal of Symbolic Computation
Extended and discretized formulations for the maximum clique problem
Computers and Operations Research
Semidefinite approximations for quadratic programs over orthogonal matrices
Journal of Global Optimization
A Variational Approach to Copositive Matrices
SIAM Review
Journal of Symbolic Computation
An iterative scheme for valid polynomial inequality generation in binary polynomial programming
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Second-Order Cone Relaxations for Binary Quadratic Polynomial Programs
SIAM Journal on Optimization
Semidefinite bounds for the stability number of a graph via sums of squares of polynomials
IPCO'05 Proceedings of the 11th international conference on Integer Programming and Combinatorial Optimization
An improved algorithm to test copositivity
Journal of Global Optimization
Journal of Global Optimization
Exploiting equalities in polynomial programming
Operations Research Letters
Note: On the polyhedral lift-and-project methods and the fractional stable set polytope
Discrete Optimization
Copositive and semidefinite relaxations of the quadratic assignment problem
Discrete Optimization
SIAM Journal on Optimization
Most Tensor Problems Are NP-Hard
Journal of the ACM (JACM)
A note on set-semidefinite relaxations of nonconvex quadratic programs
Journal of Global Optimization
On the computational complexity of membership problems for the completely positive cone and its dual
Computational Optimization and Applications
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Lovász and Schrijver [SIAM J. Optim., 1 (1991), pp. 166--190] showed how to formulate increasingly tight approximations of the stable set polytope of a graph by solving semidefinite programs (SDPs) of increasing size (lift-and-project method). In this paper we present a similar idea. We show how the stability number can be computed as the solution of a conic linear program (LP) over the cone of copositive matrices. Subsequently, we show how to approximate the copositive cone ever more closely via a hierarchy of linear or semidefinite programs of increasing size (liftings). The latter idea is based on recent work by Parrilo [Structured Semidefinite Programs and Semi-algebraic Geometry Methods in Robustness and Optimization, Ph. D. thesis, California Institute of Technology, Pasadena, CA, 2000]. In this way we can compute the stability number $\alpha(G)$ of any graph $G(V,E)$ after at most $\alpha(G)^2$ successive liftings for the LP-based approximations. One can compare this to the $n - \alpha(G)-1$ bound for the LP-based lift-and-project scheme of Lovász and Schrijver. Our approach therefore requires fewer liftings for families of graphs where $\alpha(G) IEEE Trans. Inform. Theory, 25 (1979), pp. 425--429]. We further show that the second approximation is tight for complements of triangle-free graphs and for odd cycles.