Lagrangian relaxation of quadratic matrix constraints

Quadratically constrained quadratic programs (QQPs) play an important model ing role for many diverse problems. These problems are in general NP hard a nd numerically intractable. Lagrangian relaxations often provide good appro ximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations. For several special cases of QQP, e.g., convex programs and trust region su bproblems, the Lagrangian relaxation provides the exact optimal value, i.e. , there is a zero duality gap. However, this is not true for the general QQ P, or even the QQP with two convex constraints, but a nonconvex objective. In this paper we consider a certain QQP where the quadratic constraints cor respond to the matrix orthogonality condition XXT = I. For this problem we show that the Lagrangian dual based on relaxing the constraints XXT = I and the seemingly redundant constraints (XX)-X-T = I has a zero duality gap. T his result has natural applications to quadratic assignment and graph parti tioning problems, as well as the problem of minimizing the weighted sum of the largest eigenvalues of a matrix. We also show that the technique of rel axing quadratic matrix constraints can be used to obtain a strengthened sem idefinite relaxation for the max-cut problem.