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A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. (English) Zbl 1135.90034
Summary: Existing global optimization techniques for nonconvex quadratic programming (QP) branch by recursively partitioning the convex feasible set and thus generate an infinite number of branch-and-bound nodes. An open question of theoretical interest is how to develop a finite branch-and-bound algorithm for nonconvex QP. One idea, which guarantees a finite number of branching decisions, is to enforce the first-order Karush-Kuhn-Tucker (KKT) conditions through branching. In addition, such an approach naturally yields linear programming (LP) relaxations at each node. However, the LP relaxations are unbounded, a fact that precludes their use. In this paper, we propose and study semidefinite programming relaxations, which are bounded and hence suitable for use with finite KKT-branching. Computational results demonstrate the practical effectiveness of the method, with a particular highlight being that only a small number of nodes are required.

MSC:
90C20 Quadratic programming
90C26 Nonconvex programming, global optimization
90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
Software:
BARON; NewtonKKTqp; CPLEX
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