Strong duality for a trust-region type relaxation of the quadratic assignment problem

Abstract

Lagrangian duality underlies many efficient algorithms for convex minimization problems. A key ingredient is strong duality. Lagrangian relaxation also provides lower bounds for non-convex problems, where the quality of the lower bound depends on the duality gap. Quadratically constrained quadratic programs (QQPs) provide important examples of non-convex 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 sub-problem. Surprisingly, there are classes of more complex, non-convex 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.