Enrico Miglierina

Scalarization and stability in vector optimization

A class of scalarizations of vector optimization problems is studied in order to characterize weakly efficient, efficient, and properly efficient points of a nonconvex vector problem. A parallelism is established between the different solutions of the scalarized problem and the various efficient frontiers. In particular, properly efficient points correspond to stable solutions with respect to suitable perturbations of the feasible set.

Stability for convex vector optimization problems

This article deals with the convergence (in the sense of Kuratowski–Painlevé) of the set of the minimal points of An to the set of minimal points of A, whenever {An } is a sequence of closed convex subsets of an Euclidean space, converging in the same sense to the set A. Next, we consider the convex vector optimization problem under the assumption that the objective function f is such that all its sublevel sets, restricted to the feasible region, are bounded. For this problem we investigate the convergence of the solution sets of perturbed (with respect to the feasible region and the objective function) problems both in the image space and in the decision space. We consider also the same topics for a linear problem. Finally, we apply our results to the study of stability for a vector programming problem with convex inequality and linear equality constraints.

Well-posedness and convexity in vector optimization

Abstract.We study a notion of well-posedness in vector optimization through the behaviour of minimizing sequences of sets, defined in terms of Hausdorff set-convergence. We show that the notion of strict efficiency is related to the notion of well-posedness. Using the obtained results we identify a class of well-posed vector optimization problems: the convex problems with compact efficient frontiers.

Scalarization in set optimization with solid and nonsolid ordering cones

This paper focuses on characterizations via scalarization of several kinds of minimal solutions of set-valued optimization problems, where the objective values are compared through relations between sets (set optimization). For this aim we follow an axiomatic approach based on general order representation and order preservation properties, which works in any abstract set ordered by a quasi order (i.e., reflexive and transitive) relation. Then, following this approach, we study a recent Gerstewitz scalarization mapping for set-valued optimization problems with

Characterization of solutions of multiobjective optimization problem

K

Rethinking polyhedrality for Lindenstrauss spaces

K-proper sets and a solid ordering cone

\ell_1

K