Singular Continuation: Generating Piece-wise Linear Approximations to Pareto Sets via Global Analysis

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 superposition 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 examples.Comment: 29 pages, 12 figure

Lower Convergence of Minimal Sets in Star-Shaped Vector Optimization Problems

Let An be a sequence of nonempty star-shaped sets. By using generalized domination property, we study the lower convergence of minimal sets Min An. The distinguishing feature of our results lies in disuse of convexity assumptions (only using star-shapedness)

Convergence of minimal sets in convex vector optimization

We study the behavior of the minimal sets of a sequence of convex sets $\left\{ A_{n}\right\}$ converging to a given set $A.$ The main feature of the present work is the use of convexity properties of the sets $A_{n}$ and $A$ to obtain upper and lower convergence of the minimal frontiers. We emphasize that we study both Kuratowski--Painlev\ue9 convergence and Attouch--Wets convergence of minimal sets. Moreover, we prove stability results that hold in a normed linear space ordered by a general cone, in order to deal with the most common spaces ordered by their natural nonnegative orthants (e.g., $C\left( \left[ a,b\right] \right) ,$ $l^{p}$, and $L^{p}\left( \mathbb{R}\right)$ for $1\leq p\leq \infty$). We also make a comparison with the existing results related to the topics considered in our work