Journal of Global Optimization

, Volume 71, Issue 2, pp 383–405 | Cite as

Hartley properly and super nondominated solutions in vector optimization with a variable ordering structure

  • Shokouh Shahbeyk
  • Majid Soleimani-damaneh
  • Refail Kasimbeyli


In this paper, we introduce Hartley properly and super nondominated solutions in vector optimization with a variable ordering structure. We prove the connections between Benson properly nondominated, Hartley properly nondominated, and super nondominated solutions under appropriate assumptions. Moreover, we establish some necessary and sufficient conditions for newly-defined solutions invoking an augmented dual cone approach, the linear scalarization, and variational analysis tools. In addition to the theoretical results, various clarifying examples are given.


Vector optimization Variable ordering structure (VOS) Properly nondominated solution Super nondominated solution Augmented dual cone Linear scalarization Variational analysis 



The authors would like to express their gratitude to two anonymous referees and the associate editor for their helpful comments on the earlier versions of the paper. The research of the second author was in part supported by a grant from the Iran National Science Foundation (INSF) (No. 95849588).


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Shokouh Shahbeyk
    • 1
  • Majid Soleimani-damaneh
    • 1
  • Refail Kasimbeyli
    • 2
  1. 1.School of Mathematics, Statistics and Computer Science, College of ScienceUniversity of TehranTehranIran
  2. 2.Department of Industrial EngineeringAnadolu UniversityEskisehirTurkey

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