Characterizations of set order relations and constrained set optimization problems via oriented distance function

Abstract

Set-valued optimization problems are important and fascinating field of optimization theory and widely applied to image processing, viability theory, optimal control and mathematical economics. There are two types of criteria of solutions for the set-valued optimization problems: the vector criterion and the set criterion. In this paper, we adopt the set criterion to study the optimality conditions of constrained set-valued optimization problems. We first present some characterizations of various set order relations using the classical oriented distance function without involving the nonempty interior assumption on the ordered cones. Then using the characterizations of set order relations, necessary and sufficient conditions are derived for four types of optimal solutions of constrained set optimization problem with respect to the set order relations. Finally, the image space analysis is employed to study the c-optimal solution of constrained set optimization problems, and then optimality conditions and an alternative result for the constrained set optimization problem are established by the classical oriented distance function.

Keywords:

• Constrained set optimization problems
• set order relations
• oriented distance function
• optimality conditions
• image space analysis

AMS Subject Classifications:

• 49J53
• 65K10
• 90C29

Acknowledgements

The authors are grateful to the handling editor and the two anonymous referees for their valuable comments and suggestions, which have helped to improve the previous version of the paper. Authors are also thankful to Professors Andreas H. Hamel, Elisabeth Köbis and Markus A. Köbis for their valuable discussion and suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was partially supported by the MOST [grant number 105-2221-E-039-009-MY3], the Natural Science Foundation of China [grant number 11401487], the Fundamental Research Funds for the Central Universities [grant number XDJK2014C073], the Basic and Advanced Research Project of Chongqing (cstc2016jcyjA0239), the China Postdoctoral Science Foundation [grant number 2015M582512] and the Chinese Government Scholarship Fund. The research part of the second author was done during his visit to King Fahd University of Petroleum & Minerals (KFUPM), Dhahran Saudi Arabia, where he received excellent research facilities.