Penalty methods in optimization over the weakly efficient set

The paper deals with the problem of minimizing a scalar valued function over the weakly efficient (Pareto) set associated with a multicriteria minimization problem. This problem of post-Pareto analysis has several practical applications. Its main difficulty is due to the implicit form of the weakly efficient set and to the fact that, even in the linear case, the problem is not convex. Mathematically this problem is classified as a difficult global optimization problem. To solve it we use a penalty method. For the particular case of a linear vector problem and concave scalar objective, and also for the special case of linear functional, we propose an exact penalty algorithm which finds an optimal solution in a finite number of iterations.