On the sensitivity of Pareto efficiency in set-valued optimization problems
- PDF / 341,869 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 30 Downloads / 225 Views
On the sensitivity of Pareto efficiency in set-valued optimization problems Marius Durea1,2 · Radu Strugariu3 Received: 10 June 2019 / Accepted: 13 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper we present two main situations when the limit of Pareto minima of a sequence of perturbations of a set-valued map F is a critical point of F. The concept of criticality is understood in the Fermat generalized sense by means of limiting (Mordukhovich) coderivative. Firstly, we consider perturbations of enlargement type which, in particular, cover the case of perturbation with dilating cones. Secondly, we present the case of Aubin type perturbations, and for this we introduce and study a new concept of openness with respect to a cone. Keywords Pareto minimality · Sensitivity · Criticality · Openness with respect to a cone Mathematics Subject Classification 49J53 · 90C29 · 49K40
1 Introduction and notation The aim of this work is to investigate some sensitivity issues related to optimization problems with set-valued maps. In fact, this paper has its origin in a question that recurrently appears in the literature dedicated to stability properties of various types of efficiency in vector optimization. More precisely, given a vector optimization problem, it is well known that the limit of a convergent sequence of genuine Pareto minima for some approximate problems fails to be a Pareto minimum for the initial problem (see, for instance, [26]). For this reason there are multiple attempts to get such assertions for different kinds of Pareto minimality, such as approximate minima, Geoffrion minima, and weak minima (see [4,10,15,20,24], and
B
Radu Strugariu [email protected] Marius Durea [email protected]
1
Faculty of Mathematics, “Alexandru Ioan Cuza” University, Bd. Carol I, nr. 11, 700506 Ia¸si, Romania
2
“Octav Mayer” Institute of Mathematics of the Romanian Academy, Ia¸si, Romania
3
Department of Mathematics, “Gheorghe Asachi” Technical University, Bd. Carol I, nr. 11, 700506 Ia¸si, Romania
123
Journal of Global Optimization
the references therein). On the same line of research, stability of different types of Karush– Kuhn–Tucker (KKT, for short) points have been established (see [8,9,16], and [22]). The main point we work on in this paper is to show that under some assumptions we can assure that, even the limit of a sequence of Pareto minima is not a Pareto minimum, it is, however, a critical point (when the problem is without constraints), or a KKT point (for constrained problems). This conclusion is important since, on one hand, the study of approximate problems is intended for numerical analysis of the underlying problem and, on the other hand, a numerical study of a given problem is looking exactly for critical (KKT, respectively) points, rather than for genuine solutions (see [23]). The procedure we use consists in the study of openness of a sequence of perturbation mappings in relation to the openness of the perturbed mapping and the appropriate applicatio
Data Loading...
