On Set Containment Characterizations for Sets Described by Set-Valued Maps with Applications

  • Nithirat Sisarat
  • Rabian WangkeereeEmail author
  • Gue Myung Lee


In this paper, dual characterizations of the containment of two sets involving convex set-valued maps are investigated. These results are expressed in terms of the epigraph of a conjugate function of infima associated with corresponding set-valued maps. As an application, we establish characterizations of weak and proper efficient solutions of set-valued optimization problems in the sense of vector criteria.


Set containment Convexity of set-valued maps Set-valued optimization Optimality conditions 

Mathematics Subject Classification

54C60 52A41 49J52 90C46 



The authors are extremely thankful to the anonymous referees and the editor for providing several suggestions to improve the paper to its current form greatly. This research was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0026/2555), the Thailand Research Fund, Grant No. RSA6080077, and Naresuan University. The third author was supported by the National Research Foundation of Korea (NRF) Grant funded by Korea government (MSIT) (NRF-2017R1E1A1A03069931).


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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceNaresuan UniversityPhitsanulokThailand
  2. 2.Research center for Academic Excellence in MathematicsNaresuan UniversityPhitsanulokThailand
  3. 3.Department of Applied MathematicsPukyong National UniversityBusanKorea

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