Hierarchical stratification of Pareto sets

Abstract

In smooth and convex multiobjective optimization problems the set of Pareto optima is diffeomorphic to an $m-1$ dimensional simplex, where $m$ is the number of objective functions. The vertices of the simplex are the optima of the individual functions and the $(k-1)$-dimensional facets are the Pareto optimal set of $k$ functions subproblems. Such a hierarchy of submanifolds is a geometrical object called stratification and the union of such manifolds, in this case the set of Pareto optima, is called a stratified set. We discuss how these geometrical structures generalize in the non convex cases, we survey the known results and deduce possible suggestions for the design of dedicated optimization strategies.