A class of optimization problems over the efficient set of a multiple criteria nonlinear programming problem

One of the most important and interesting apporaches in multiple criteria optimization is the problem of optimizing some function over the set of efficient solutions. This is a very difficult multiextremal global optimization problem, even for the case that the underlying multiple criteria program is linear and the function to be optimized over the efficient set is linear as well. In this article, we consider the problem of maximizing a special function over the efficient set of a multiple criteria concave maximization problem under the assumption that the objective function is a nondecreasing composite quasiconvex function of the concave criteria. We show that this problem can be formulated as a special global optimization problem in the outcome space. An algorithm of the outer approximation type is proposed for solving the resulting problem. Preliminary computational experiments show that this algorithm can work well for problems with a small number of criteria (less than 8), while the dimension of the decision space can be fairly large.