Semidefinite programming for discrete optimization and matrix completion problems
Discrete Applied Mathematics
Image and Vision Computing
Second order cone programming relaxation for quadratic assignment problems
Optimization Methods & Software
Journal of Global Optimization
A Low-Dimensional Semidefinite Relaxation for the Quadratic Assignment Problem
Mathematics of Operations Research
Semidefinite approximations for quadratic programs over orthogonal matrices
Journal of Global Optimization
On Equivalence of Semidefinite Relaxations for Quadratic Matrix Programming
Mathematics of Operations Research
Quadratic programs over the Stiefel manifold
Operations Research Letters
Copositive and semidefinite relaxations of the quadratic assignment problem
Discrete Optimization
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Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate 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 subproblems, the Lagrangian relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective. In this paper we consider a certain QQP where the quadratic constraints correspond 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 XT X=I has a zero duality gap. This result has natural applications to quadratic assignment and graph partitioning 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 relaxing quadratic matrix constraints can be used to obtain a strengthened semidefinite relaxation for the max-cut problem.