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We consider a quadratic programming problem with quadratic cone constraints and an additional geometric constraint. Under suitable assumptions, we establish necessary and sufficient conditions for optimality of a KKT point and, in particular, we characterize optimality by using strong duality as a regularity condition. We consider in details the case where the feasible set is defined by two quadratic equality constraints and, finally, we analyse simultaneous diagonalizable quadratic problems, where the Hessian matrices of the involved quadratic functions are all diagonalizable by means of the same orthonormal matrix.

First- and Second-Order Optimality Conditions for Quadratically Constrained Quadratic Programming Problems