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Second-order optimality conditions for nonconvex set-constrained optimization problems. (English) Zbl 1508.90067

This paper deals with the second-order optimality condition for the general nonconvex set-constrained optimization problem. It is well known that the second-order optimality condition must involve in some way the second-order tangent set and the nonconvexity of this second-order tangent set is also an issue. The authors consider two different approaches for handling the nonconvex second-order tangent sets. In the first approach, the concept of lower generalized support functions is introduced in order to state the second-order necessary optimality conditions. In the second approach, the concept of a directional regular tangent cone is introduced. It is shown that these second-order conditions are equivalent with primal second-order conditions and are, in general, stronger than the one obtained with the first approach. Second-order sufficient conditions for optimality is discussed. Further, examples that illustrate the second-order necessary and sufficient conditions are presented.

MSC:

90C26 Nonconvex programming, global optimization
90C46 Optimality conditions and duality in mathematical programming
49J53 Set-valued and variational analysis
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