First order optimality conditions in set-valued optimization

Crespi Giovanni P., Ginchev Ivan () and Rocca Matteo ()
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Crespi Giovanni P.: Department of Economics, University of Insubria, Italy
Ginchev Ivan: Department of Mathematics Varna, Bulgaria
Rocca Matteo: Department of Economics, University of Insubria, Italy

Economics and Quantitative Methods from Department of Economics, University of Insubria

Abstract: A a set-valued optimization problem minC F(x), x 2 X0, is considered, where X0 X, X and Y are Banach spaces, F : X0 Y is a set-valued function and C Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x0, y0), y0 2 F(x0), and are called minimizers. In particular the notions of w-minimizer (weakly efficient points), p-minimizer (properly efficient points) and i-minimizer (isolated minimizers) are introduced and their characterization in terms of the so called oriented distance is given. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive first order conditions, that is conditions in terms of suitable first order derivatives of F, for a pair (x0, y0), where x0 2 X0, y0 2 F(x0), to be a solution of this problem. We define and apply for this purpose the directional Dini derivative. Necessary conditions and sufficient conditions a pair (x0, y0) to be a w-minimizer, and similarly to be a i-minimizer are obtained. The role of the i-minimizers, which seems to be a new concept in set-valued optimization, is underlined. For the case of w-minimizers some comparison with existing results is done. Key words: Vector optimization, Set-valued optimization, First-order optimality conditions.

Pages: 19 pages
Date: 2004-06
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Citations: View citations in EconPapers (1)

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