A note on lack of strong duality for quadratic problems with orthogonal constraints

The general quadratically constrained quadratic program (QQP) is an important modelling tool for many diverse problems. The QQP is 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 and can be solved efficiently. For several special cases of QQP, the Lagrangian relaxation provides the exact optimal value. This means that there is a zero duality gap and the problem is tractable. It is important to know for which cases this is true, since they can then be used as subproblems to improve Lagrangian relaxation for intractable QQPs. In this paper we study the special QQP with orthogonal (matrix) constraints XX^T=I. If C=0, the zero duality gap result holds if the redundant orthogonal constraints X^TX=I are added. We show that this is not true in the general case. However, we show how to close the duality gap in the pure linear case by adding variables in addition to constraints.