On approximate optimality conditions for robust mufti-objective convex optimization problems

In this paper, we are interested in the study of approximate optimality conditions for weakly epsilon-efficient solutions to robust multi-objective optimization problems ((RMOP) for short) in view of its associated minimax optimization problem (MMOP). To this end, we first establish the relationship between a weakly epsilon-efficient solution to the problem (RMOP) and an a-solution to the problem (MMOP), where epsilon = (epsilon(1), ..., epsilon(p)) is an element of R-+(p) \ {0} and alpha = max(j=1, ..., p{epsilon j}). Then, we explore the representation of the so-called beta-normal set (where beta >= 0 is a given parameter) to a closed convex set at some reference point by two methods. At last, by employing the alpha-subdifferential of the max-function and the obtained representation of the beta-normal set, we establish an approximate necessary optimality condition for the problem (RMOP). Moreover, we also give an example to illustrate our results.