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On Optimization Problems with Set-Valued Objective Maps: Existence and Optimality

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Abstract

In this paper, we consider constrained optimization problems with set-valued objective maps. First, we define three types of quasi orderings on the set of all non-empty subsets of n-dimensional Euclidean space. Second, by using these quasi orderings, we define the concepts of lower semi-continuity for set-valued maps and investigate their properties. Finally, based on these results, we define the concepts of optimal solutions to constrained optimization problems with set-valued objective maps and we give some conditions under which these optimal solutions exist to the problems and give necessary and sufficient conditions for optimality.

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Authors and Affiliations

  1. Faculty of Economics, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa, Japan, 920-1192

    Takashi Maeda

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  1. Takashi Maeda
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Correspondence to Takashi Maeda.

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Communicated by Jafar Zafarani.

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Maeda, T. On Optimization Problems with Set-Valued Objective Maps: Existence and Optimality. J Optim Theory Appl 153, 263–279 (2012). https://doi.org/10.1007/s10957-011-9952-x

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  • DOI: https://doi.org/10.1007/s10957-011-9952-x

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