Approximation of the Stability Number of a Graph via Copositive Programming
SIAM Journal on Optimization
On Copositive Programming and Standard Quadratic Optimization Problems
Journal of Global Optimization
Branch-and-bound approaches to standard quadratic optimization problems
Journal of Global Optimization
Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming
Journal of Global Optimization
A Variational Approach to Copositive Matrices
SIAM Review
Journal of Global Optimization
Journal of Global Optimization
Tightening a copositive relaxation for standard quadratic optimization problems
Computational Optimization and Applications
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The standard quadratic program (QPS) is minx驴驴xTQx, where $$\Delta\subset\Re^n$$ is the simplex 驴 = {x 驴 0 驴 驴i=1n xi = 1}. QPS can be used to formulate combinatorial problems such as the maximum stable set problem, and also arises in global optimization algorithms for general quadratic programming when the search space is partitioned using simplices. One class of `d.c.' (for `difference between convex') bounds for QPS is based on writing Q=S驴T, where S and T are both positive semidefinite, and bounding xT Sx (convex on 驴) and 驴xTx (concave on 驴) separately. We show that the maximum possible such bound can be obtained by solving a semidefinite programming (SDP) problem. The dual of this SDP problem corresponds to adding a simple constraint to the well-known Shor relaxation of QPS. We show that the max d.c. bound is dominated by another known bound based on a copositive relaxation of QPS, also obtainable via SDP at comparable computational expense. We also discuss extensions of the d.c. bound to more general quadratic programming problems. For the application of QPS to bounding the stability number of a graph, we use a novel formulation of the Lovasz 驴 number to compare 驴, Schrijver's 驴驴, and the max d.c. bound.