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Semidefinite approximations for quadratic programs over orthogonal matrices. (English) Zbl 1203.90121
Summary: Finding global optimum of a non-convex quadratic function is in general a very difficult task even when the feasible set is a polyhedron. We show that when the feasible set of a quadratic problem consists of orthogonal matrices from \({\mathbb{R}^{n\times k}}\) , then we can transform it into a semidefinite program in matrices of order \(kn\) which has the same optimal value. This opens new possibilities to get good lower bounds for several problems from combinatorial optimization, like the Graph partitioning problem (GPP), the Quadratic assignment problem (QAP) etc. In particular we show how to improve significantly the well-known Donath-Hoffman eigenvalue lower bound for GPP by semidefinite programming. In the last part of the paper we show that the copositive strengthening of the semidefinite lower bounds for GPP and QAP yields the exact values.

90C20 Quadratic programming
90C22 Semidefinite programming
90C26 Nonconvex programming, global optimization
90C27 Combinatorial optimization
Full Text: DOI
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