Set contraction algorithm for computing Pareto set in nonconvex nonsmooth multiobjective optimization

Nonconvex nonsmooth multiobjective programs min f(i)(x), x is an element of X, i = 1,...,k, are studied 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 f(i)(x) are continuous over X, then a non-Pareto set, if nonempty, has nonempty interior in the topology of X. If (x) over bar is an element of X is not a Pareto point, then a set N-*((x) over bar) is constructed which is robust, contains (x) over bar, 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. (C) 2004 Elsevier Ltd. All rights reserved.