Equivalent Sufficient Conditions for Global Optimality of Quadratically Constrained Quadratic Program

Abstract: We study the equivalence of several well-known sufficient optimality conditions for a general quadratically constrained quadratic program (QCQP). The conditions are classified in two categories. The first one is for determining an optimal solution and the second one is for finding an optimal value. The first category of conditions includes the existence of a saddle point of the Lagrangian function and the existence of a rank-1 optimal solution of the primal SDP relaxation of QCQP. The second category includes $\eta_p = \zeta$, $\eta_d = \zeta$, and $\varphi = \zeta$, where $\zeta$, $\eta_p$, $\eta_d$, and $\varphi$ denote the optimal values of QCQP, the dual SDP relaxation, the primal SDP relaxation and the Lagrangian dual, respectively. We show the equivalence of these conditions with or without the existence of an optimal solution of QCQP and/or the Slater constraint qualification for the primal SDP relaxation. The results on the conditions are also extended to the doubly nonnegative relaxation of equality constrained QCQP in nonnegative variables.