Higher-order conditions for strict efficiency

Abstract

Luu and Kien (On higher order conditions for strict efficiency, Soochow J. Math. 33 (2007), pp. 17–31), proposed higher-order conditions for strict efficiency of vector optimization problems based on the derivatives introduced in Ginchev (Higher order optimality conditions in nonsmooth optimization, Optimization 51 (2002), pp. 47–72). These derivatives are defined for scalar functions and in their terms necessary and sufficient conditions are obtained a point to be strictly efficient (isolated) minimizer of a given order for quite arbitrary scalar function. Passing to vector functions, Luu and Kien lose the peculiarity that the optimality conditions work with arbitrary functions. In this article, applying the mentioned derivatives for the scalarized problem and restoring the original idea, optimality conditions for strict efficiency of a given order are proposed, which work with quite arbitrary vector functions. It is shown that the results of Luu and Kien are corollaries of the given conditions and generalizations are discussed.