Strong Duality for a Trust-Region Type Relaxation of the Quadratic Assignment Problem

Kurt Anstreicher, Xin Chen, Henry Wolkowicz, Ya-Xiang Yuan

Lagrangian duality underlies many efficient algorithms for convex minimization problems. A key ingredient is strong duality. Lagrangian relaxation also provides lower bounds for nonconvex problems, where the quality of the lower bound depends on the duality gap. Quadratically constrained quadratic programs (QQPs) provide important examples of nonconvex programs. For the simple case of one quadratic constraint (the trust region subproblem) strong duality holds. In addition, necessary and sufficient (strengthened) second order optimality conditions exist. However, these duality results already fail for the two trust region subproblem. Surprisingly, there are classes of more complex, nonconvex QQPs where strong duality holds. One example is the special case of orthogonality constraints, which arise naturally in relaxations for the quadratic assignment problem (QAP). In this paper we show that strong duality also holds for a relaxation of QAP where the orthogonality constraint is replaced by a semidefinite inequality constraint. Using this strong duality result, and semidefinite duality, we develop new trust-region type necessary and sufficient optimality conditions for these problems. Our proof of strong duality introduces and uses a generalization of the Hoffman-Wielandt inequality.

Research Report CORR 98-31
University of Waterloo
Department of Combinatorics and Optimization
Waterloo, Ontario N2L 3G1, Canada