Solving Optimization Problems over the Weakly Efficient Set

In this paper, we study the optimization problem (PWE) of minimizing a convex function over the set of weakly efficient solutions of a convex multiobjective problem. This is done by using the fact that each lower semicontinuous convex function is an upper envelope of its affine minorants together with a generalized cutting plane method. We give necessary conditions for optimal solutions of the problem (PWE). Moreover, a novel algorithm for solving the problem (PWE) together with numerical results are presented. We also prove that the proposed algorithm terminates after a finite numbers of iterations, and the algorithm is coded in MATLAB language and evaluated by numerical examples.