Stochastic Optimization over a Pareto Set Associated with a Stochastic Multi-Objective Optimization Problem

We deal with the problem of minimizing the expectation of a real valued random function over the weakly Pareto or Pareto set associated with a Stochastic MultiObjective Optimization Problem (SMOP) whose objectives are expectations of random functions. Assuming that the closed form of these expectations is difficult to obtain, we apply the Sample Average Approximation method (SAA-N, where N is the sample size) in order to approach this problem. We prove that the Hausdorff-Pompeiu distance between the SAA-N weakly Pareto sets and the true weakly Pareto set converges to zero almost surely as N goes to infinity, assuming that all the objectives of our (SMOP) are strictly convex. Then we show that every cluster point of any sequence of SAA-N optimal solutions (N=1,2,. . . ) is almost surely a true optimal solution. To handle also the nonconvex case, we assume that the real objective to be minimized over the Pareto set depends on the expectations of the objectives of the (SMOP), i.e. we optimize over the outcome space of the (SMOP). Then, whithout any convexity hypothesis, we obtain the same type of results for the Pareto sets in the outcome spaces. Thus we show that the sequence of SAA-N optimal values (N=1,2 ...) converges almost surely to the true optimal value.