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Scalarizations and Optimality of Constrained Set-Valued Optimization Using Improvement Sets and Image Space Analysis

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Abstract

In this paper, we aim at applying improvement sets and image space analysis to investigate scalarizations and optimality conditions of the constrained set-valued optimization problem. Firstly, we use the improvement set to introduce a new class of generalized convex set-valued maps. Secondly, under suitable assumptions, some scalarization results of the constrained set-valued optimization problem are obtained in the sense of (weak) optimal solution characterized by the improvement set. Finally, by considering two classes of nonlinear separation functions, we present the separation between two suitable sets in image space and derive some optimality conditions for the constrained set-valued optimization problem. It shows that the existence of a nonlinear separation is equivalent to a saddle point condition of the generalized Lagrangian set-valued functions.

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Acknowledgements

This work was supported by the National Nature Science Foundation of China (11431004, 11861002) and the Key Project of Chongqing Frontier and Applied Foundation Research (cstc2018jcyj-yszxX0009, cstc2017jcyjBX0055, cstc2015jcyjBX0113).

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Authors and Affiliations

  1. College of Sciences, Chongqing University of Technology, Chongqing, 400054, China

    Zhiang Zhou

  2. College of Mathematics, Sichuan University, Chengdu, 610065, China

    Wang Chen

  3. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 400047, China

    Xinmin Yang

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  1. Zhiang Zhou
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  2. Wang Chen
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  3. Xinmin Yang
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Correspondence to Zhiang Zhou.

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Jafar Zafarani.

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Zhou, Z., Chen, W. & Yang, X. Scalarizations and Optimality of Constrained Set-Valued Optimization Using Improvement Sets and Image Space Analysis. J Optim Theory Appl 183, 944–962 (2019). https://doi.org/10.1007/s10957-019-01554-3

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  • DOI: https://doi.org/10.1007/s10957-019-01554-3

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