Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions

The unconstrained binary quadratic programming problem: a survey

“…They derive a sufficient optimality condition which guarantees that a given feasible point is a global optimal for UBQP as well as a necessary global optimality condition. Jeyakumar, Rubinov, and Wu (2007) given that allow instances of UBQP to be converted into smooth concave maximization dual problems over a closed convex feasible region without a duality gap. Finally, the relationship between canonical duality theory and semi-definite programming for UBQP is discussed.…”

Necessary and sufficient global optimality conditions for NLP reformulations of linear SDP problems

“…These conditions extend the necessary and sufficient ones proved in [5] for the special case of the semidefinite relaxation of the max cut problem and can be related to the necessary and sufficient global optimality conditions established for some classes of nonconvex quadratic problems (see e.g. [6,11,9]). …”

“…We point out once more that problem (NLP r ) is a non convex optimization problem so that necessary and sufficient global optimality conditions are usually not available. In general, for non convex quadratic-quadratic minimization problems (quadratic objective function and quadratic constraints) global optimality conditions can be found in some particular situations (see [6,11,9]). A well known case is when there is one single quadratic constraint (equality or inequality), that is the case of the generalized trust region problem [11].…”

On semidefinite bounds for maximization of a non-convex quadratic objective over thel1unit ball

“…Since the SDP relaxation method is not the only approach to tackle optimization problems with quadratic objective and constraint functions, we mention here the work on global optimality conditions for non-convex optimization problems developed in a series of papers by Hiriart-Urruty, see e.g., [2][3][4] and Jeyakumar et al [6,7]. Hiriart-Urruty develops a general global optimality condition based on a generalized subdifferential concept, and specializes the condition to several problems of non-convex optimization, including maximization of a convex quadratic function subject to strictly convex quadratic inequalities, minimization of a quadratic function subject to a single quadratic inequality (trust-region problem), and subject to two quadratic inequalities (two-trust-region problem).…”