Second‐order conditions for nonsmooth multiobjective optimization problems with inclusion constraints

This paper investigates second‐order optimality conditions for general multiobjective optimization problems with constraint set‐valued mappings and an arbitrary constraint set in Banach spaces. Without differentiability nor convexity on the data and with a metric regularity assumption the second‐order necessary conditions for weakly efficient solutions are given in the primal form. Under some additional assumptions and with the help of Robinson ‐Ursescu open mapping theorem we obtain dual second‐order necessary optimality conditions in terms of Lagrange‐Kuhn‐Tucker multipliers. Also, the second‐order sufficient conditions are established whenever the decision space is finite dimensional. To this aim, we use the second‐order projective derivatives associated to the second‐order projective tangent sets to the graphs introduced by Penot. From the results obtained in this paper, we deduce and extend, in the special case some known results in scalar optimization and improve substantially the few results known in vector case.