First-order optimality conditions in set-valued optimization |
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Authors: | Giovanni P Crespi Ivan Ginchev Matteo Rocca |
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Institution: | (1) Facoltà di Scienze Economiche, Université de la Vallé d’Aoste, Via Duca Degli Abruzzi 4, 11100 Aosta, Italy;(2) Department of Mathematics, Technical University of Varna, Studentska Str. 1, 9010 Varna, Bulgaria;(3) Department of Economics, University of Insubria, Via Monte Generoso 71, 21100 Varese, Italy |
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Abstract: | A a set-valued optimization problem min
C
F(x), x ∈X
0, is considered, where X
0 ⊂ X, X and Y are normed spaces, F: X
0 ⊂ Y is a set-valued function and C ⊂ Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x
0,y
0), y
0 ∈F(x
0), and are called minimizers. The notions of w-minimizers (weakly efficient points), p-minimizers (properly efficient points) and i-minimizers (isolated minimizers) are introduced and characterized through the so called oriented distance. The relation between
p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive in terms of the Dini
directional derivative first order necessary conditions and sufficient conditions a pair (x
0, y
0) to be a w-minimizer, and similarly to be a i-minimizer. The i-minimizers seem to be a new concept in set-valued optimization. For the case of w-minimizers some comparison with existing results is done. |
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Keywords: | Vector optimization Set-valued optimization First-order optimality conditions |
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