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On Tucker-Type Alternative Theorems and Necessary Optimality Conditions for Nonsmooth Multiobjective Optimization

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Abstract

This paper concentrates on necessary conditions for properly efficient solutions in nonsmooth multiobjective optimization problems. We first present a generalization of Tucker’s alternative theorem for conic nonlinear systems, provided that a closedness condition holds. Some sufficient conditions for the validity of such a closedness condition are given. As applications, under the weak Abadie regularity condition, we then establish the primal and the strong Karush/Kuhn–Tucker (dual) necessary optimality conditions for an efficient solution to be locally properly efficient in Borwein’s sense. The primal and the dual conditions are formulated as an equivalent pair by means of the Tucker-type alternative results. Finally we give an example to illustrate that Borwein’s locally properly efficient solution cannot be reduced to the only efficient one in the main results.

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Acknowledgements

The authors would like to thank Prof. Fabian Flores-Bazán and two anonymous referees whose detailed comments and constructive suggestions, including the remark for adding Sect. 4 on discussing optimality results of an infinite dimensional vector problem, which helped to improve the paper. This research was supported by the National Natural Science Foundation of China (Grant Number: 11971078), the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant Number: KJQN202000740) and the Joint Training Base Construction Project for Graduate Students in Chongqing (Grant Number: JDLHPYJD2021016).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, China

    Min Feng

  2. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Shengjie Li & Jie Wang

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  1. Min Feng
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  2. Shengjie Li
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  3. Jie Wang
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Correspondence to Min Feng.

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Communicated by Fabian Flores-Bazán.

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Feng, M., Li, S. & Wang, J. On Tucker-Type Alternative Theorems and Necessary Optimality Conditions for Nonsmooth Multiobjective Optimization. J Optim Theory Appl 195, 480–503 (2022). https://doi.org/10.1007/s10957-022-02092-1

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