Higher-Order Efficiency Conditions for Vector Nonsmooth Optimization Problems Using the Higher-Order Gâteaux Derivatives

In this article, we investigate the higher-order nonsmooth optimality conditions for vector optimization problems with inequality, equality and set constraints in terms of the higher-order G & acirc;teaux derivatives. First, we propose various higher-order Mangasarian-Fromovitz nonsmooth constraint qualifications for such problems. Second, we formulate higher-order KKT-type necessary optimality conditions for the local weak efficient solutions of the nonsmooth vector equilibrium problem with constraints (CVEP) and its special cases. An application of the result to the resources assignment problem with set, inequality, equality constraints is derived. Under some suitable assumptions involving a set constraint, the higher-order nonsmooth necessary optimality conditions become the higher-order sufficient optimality conditions via the higher-order directional/G & acirc;teaux derivatives.