Non-smooth Optimality for Robust Multi-objective Optimization Problems

This article is concerned with non-smooth/nonconvex robust multi-objective optimization problems involving uncertain inequality and equality constraints. Employing some advanced tools of variational analysis such as the approximate extremal principle and the weak fuzzy sum rule for the Frechet subdifferential, we first drive a fuzzy necessary optimality condition of a non-smooth/nonconvex multi-objective optimization problem without any constrained qualification in the sense of the Frechet subdifferential. Then by exploiting the obtained fuzzy optimality condition, the non-smooth version of Fermat’s rule and formulae for the limiting subdifferential of an infinite family of non-smooth functions, we establish a necessary optimality condition in terms of the limiting subdifferential for weakly robust efficient solutions of the reference problem. Further,we present an example to illustrate this condition for an uncertain multi-objective optimization problem involving equality and inequality constraints.Finally sufficient conditions for weakly robust efficient solutions and robust efficient solutions of the problems are provided by presenting new concepts of generalized convexity.