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

  • Zhiang ZhouEmail author
  • Wang Chen
  • Xinmin Yang
Article
  • 9 Downloads

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.

Keywords

Image space analysis Set-valued maps Improvement set Scalarization Optimality 

Mathematics Subject Classification

90C26 90C29 90C46 26B25 

Notes

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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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of SciencesChongqing University of TechnologyChongqingChina
  2. 2.College of MathematicsSichuan UniversityChengduChina
  3. 3.School of Mathematical SciencesChongqing Normal UniversityChongqingChina

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