Set contraction algorithm for computingpareto set in nonconvex nonsmooth multiobjective optimization

Nonconvex nonsmooth multiobjective programs min fi(χ), χX, i = 1, , k, arestudied in regard to the structure of their Pareto set. Basic lemmas and theorems are proved including the result that, if a feasible set X is robust and all fi(χ) are continuous over X, then a non-Pareto set, if nonempty, has nonempty interior in the topology of X. If x ˜ ∈ X is not a Pareto point, then a set N * ( x ¯ ) is constructed which is robust, contains i, and does not contain Pareto points. On this basis, a monotonic set contraction algorithm is developed that converges onto the entire exact Pareto set, if nonempty, or yields its approximation with given precision in a finite number of iterations. Simultaneously, approximations for the ideal point and for the balance set are obtained. The method is ready for computer implementation. Illustrative examples are presented to facilitate software design.