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We study the equivalence of several well-known sufficient optimalityconditions for a general quadratically constrained quadratic program (QCQP).The conditions are classified in two categories. The first one is fordetermining an optimal solution and the second one is for finding an optimalvalue. The first category of conditions includes the existence of a saddlepoint of the Lagrangian function and the existence of a rank-1 optimal solutionof 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 SDPrelaxation, the primal SDP relaxation and the Lagrangian dual, respectively. Weshow the equivalence of these conditions with or without the existence of anoptimal solution of QCQP and/or the Slater constraint qualification for theprimal SDP relaxation. The results on the conditions are also extended to thedoubly nonnegative relaxation of equality constrained QCQP in nonnegativevariables.

Equivalent Sufficient Conditions for Global Optimality of Quadratically Constrained Quadratic Program