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ɛ-Subdifferentials of Set-valued Maps and ɛ-Weak Pareto Optimality for Multiobjective Optimization

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  • A. Taa

Abstract

In this paper we consider vector optimization problems where objective and constraints are set-valued maps. Optimality conditions in terms of Lagrange-multipliers for an ɛ-weak Pareto minimal point are established in the general case and in the case with nearly subconvexlike data. A comparison with existing results is also given. Our method used a special scalarization function, introduced in optimization by Hiriart-Urruty. Necessary and sufficient conditions for the existence of an ɛ-weak Pareto minimal point are obtained. The relation between the set of all ɛ-weak Pareto minimal points and the set of all weak Pareto minimal points is established. The ɛ-subdifferential formula of the sum of two convex functions is also extended to set-valued maps via well known results of scalar optimization. This result is applied to obtain the Karush–Kuhn–Tucker necessary conditions, for ɛ-weak Pareto minimal points Copyright Springer-Verlag 2005

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  • A. Taa, 2005. "ɛ-Subdifferentials of Set-valued Maps and ɛ-Weak Pareto Optimality for Multiobjective Optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 62(2), pages 187-209, November.
    • Handle: RePEc:spr:mathme:v:62:y:2005:i:2:p:187-209
    DOI: 10.1007/s00186-005-0007-7
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    1. J. B. Hiriart-Urruty, 1979. "Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces," Mathematics of Operations Research, INFORMS, vol. 4(1), pages 79-97, February.
    2. Z. F. Li, 1998. "Benson Proper Efficiency in the Vector Optimization of Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 98(3), pages 623-649, September.
    3. Johannes Jahn & Rüdiger Rauh, 1997. "Contingent epiderivatives and set-valued optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(2), pages 193-211, June.
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