  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.
