Second-order optimality conditions with the envelope-like effect in nonsmooth multiobjective mathematical programming I: ll-stability and set-valued directional derivatives

Second-order necessary conditions and sufficient conditions, with the envelope-like effect, for optimality in nonsmooth multiobjective mathematical programming are established. We use set-valued second-order directional derivatives and impose strict differentiability for necessary conditions and ll-stability for sufficient conditions, avoiding continuous differentiability. The results improve and sharpen several recent existing ones. Examples are provided to show advantages of our theorems over some known ones in the literature. In Part I, we consider ll-stability and second-order set-valued directional derivatives of vector functions. Part II is devoted to second-order necessary optimality conditions and sufficient ones.