Abstract
Consider a parametrized multiobjective optimization problem with parameteru. LetG(u) be the objective space image of the feasible region, and letW(u)=MinG(u) (the perturbation map) be the efficient set in the objective space. The purpose of this paper is to investigate the relationship between the contingent derivativeDW ofW with respect tou and the contingent derivativeDG ofG with respect tou. Tanino (Ref. 1) proves that MinDG⊂DW under certain conditions. In this paper, we prove that MinDG=MinDW under weaker conditions than Tanino's and that MinDG=DW under certain conditions. The paper does this by introducing a weaker notion of set-valued derivative. Along the way, the paper improves another of Tanino's results by using weaker conditions.
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References
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Communicated by P. L. Yu
The author would like to thank two anonymous referees for helpful comments. The author would also like to thank Professor P. L. Yu for encouragement and useful suggestions.
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Shi, D.S. Contingent derivative of the perturbation map in multiobjective optimization. J Optim Theory Appl 70, 385–396 (1991). https://doi.org/10.1007/BF00940634
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DOI: https://doi.org/10.1007/BF00940634