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\(\epsilon \)-Optimality conditions of vector optimization problems with set-valued maps based on the algebraic interior in real linear spaces

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Abstract

In this paper, firstly, the necessary and sufficient optimality conditions for \(\epsilon \)-global properly efficient elements of set-valued optimization problems, respectively, are established in linear spaces. Secondly, an equivalent characterization of \(\epsilon \)-global proper saddle point is presented. Finally, the necessary and sufficient conditions for \(\epsilon \)-global properly saddle point of a Lagrangian set-valued map are obtained. The results in this paper generalize some known results in the literature.

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Acknowledgments

This work was supported by the projects of the National Nature Science Foundation of China (11271391, 11171363 and 11226233), the Natural Science Foundation of Chongqing (CSTC 2011jjA00022 and CSTC 2011BA0030), the Natural Science Foundation of Zhejiang Province (LY12A01005), the Special Fund of Chongqing Key Laboratory (CSTC 2011KLORSE01), the Scientific Research Starting Foundation for Doctors of Chongqing University of Technology (2011ZD40) and the project of the third batch support program for excellent talents of Chongqing City High Colleges. The authors would like to express their thanks to two anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University of Technology, Chongqing, 400054, China

    Zhi-Ang Zhou

  2. School of Mathematics, Chongqing Normal University, Chongqing, 400047, China

    Xin-Min Yang & Jian-Wen Peng

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  1. Zhi-Ang Zhou
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  2. Xin-Min Yang
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Correspondence to Jian-Wen Peng.

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Zhou, ZA., Yang, XM. & Peng, JW. \(\epsilon \)-Optimality conditions of vector optimization problems with set-valued maps based on the algebraic interior in real linear spaces. Optim Lett 8, 1047–1061 (2014). https://doi.org/10.1007/s11590-013-0620-y

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  • DOI: https://doi.org/10.1007/s11590-013-0620-y

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