Singular continuation: Generating piecewise linear approximations to pareto sets via global analysis

Abstract. We propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale (Dynamical systems, New York, 1973, pp. 531–544). The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. The method distinguishes the hierarchy between singular set, Pareto critical set and stable Pareto critical set, and can handle the problem of super-position of local Pareto fronts, occurring in the general nonconvex case. Furthermore, a quadratic convergence result in a suitable set–wise sense is proven and tested in a number of numerical exam-ples. Key words. Multiobjective optimization, Pareto critical set, Delaunay tessellations in general dimension, stability of mappings AMS subject classifications. 90C29, 58K2