Paper's Title:

Topological Aspects of Scalarization in Vector Optimization Problems.

Author(s):

Peter I. Kogut, Rosanna Manzo and Igor V. Nechay

Department of Differential Equations,
Dnipropetrovsk National University, Naukova STR.,
 13, 49010 Dnipropetrovsk,
Ukraine
 p.kogut@i.ua

Università di Salerno,
Dipartimento di Ingegneria dell'Informazione e Matematica Applicata,
Via Ponte don Melillo, 84084 Fisciano (SA),
Italy
 manzo@diima.unisa.it

Department of Technical Cybernetics,
Dnipropetrovsk Technical University,
Acad. Lazarjan STR., 2, 49010 Dnipropetrovsk,
Ukraine
 i.nechay@i.ua

Abstract:

In this paper, we study vector optimization problems in partially ordered Banach spaces. We suppose that the objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. We derive sufficient conditions for existence of efficient solutions of the above problems and discuss the role of topological properties of the objective space. We discuss the scalarization of vector optimization problems when the objective functions are vector-valued mappings with a weakened property of lower semicontinuity. We also prove the existence of the so-called generalized efficient solutions via the scalarization process. All principal notions and assertions are illustrated by numerous examples.

Paper's Title:

Generalized Efficient Solutions to One Class of Vector Optimization Problems in Banach Space

Author(s):

Peter I. Kogut, Rosanna Manzo, and Igor V. Nechay

Department of Differential Equations,
Dnipropetrovsk National University,
Naukova str., 13, 49050 Dnipropetrovsk,
Ukraine
 p.kogut@i.ua

 Dipartimento di Ingegneria Dell’informazione e Matematica Applicata,
Università di Salerno,
Via Ponte Don Melillo, 84084 Fisciano (Sa),
Italy
manzo@diima.unisa.it

 Department of Technical Cybernetics,
Dnipropetrovsk Technical University,
Acad. Lazarjan str., 2, 49010
Dnipropetrovsk,
Ukraine
 i.nechay@i.ua

Abstract:

In this paper, we study vector optimization problems in Banach spaces for essentially nonlinear operator equations with additional control and state constraints. We assume that an objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. Using the penalization approach we derive both sufficient and necessary conditions for the existence of efficient solutions of the above problems. We also prove the existence of the so-called generalized efficient solutions via the scalarization of some penalized vector optimization problem.

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