Duality and robust duality for special nonconvex homogeneous quadratic programming under certainty and uncertainty environment

Abstract

In this paper, we discuss a kind of special nonconvex homogenous quadratic programming (HQP) and the methods to solve the HQP in an environment with certainty or uncertainty. In an environment with certainty, we first establish a strong duality between the HQP and its Lagrange dual problem, with the help of the fact that the Lagrange dual problem is equivalent to a convex semidefinite programming (SDP). Then we obtain a global solution to the HQP by solving the convex SDP. Furthermore, in an environment with uncertainty, we formulate the robust counterpart of the HQP to cope with uncertainty. We also establish the robust strong duality between the robust counterpart and its optimistic counterpart under a mild assumption. Since the counterpart is equivalent to a convex SDP under the same assumption, we can obtain a global solution to the robust counterpart by solving the convex SDP under the same assumption.