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Journal of Optimization Theory and Applications

, Volume 179, Issue 1, pp 137–162 | Cite as

Robustness in Deterministic Vector Optimization

  • Morteza Rahimi
  • Majid Soleimani-damaneh
Article

Abstract

In this paper, robust efficient solutions of a vector optimization problem, whose image space is ordered by an arbitrary ordering cone, are defined. This is done from different points of view, including set based and norm based. The relationships between these solution concepts are established. Furthermore, it is shown that, for a general vector optimization problem, each norm-based robust efficient solution is a strictly efficient solution; each isolated efficient solution is a norm-based robust efficient solution; and, under appropriate assumptions, each norm-based robust efficient solution is a Henig properly efficient solution. Various necessary and sufficient conditions for characterizing norm-based robust solutions of a general vector optimization problem, in terms of the tangent and normal cones and the nonascent directions, are presented. An optimization problem for calculating a robustness radius is provided, and then, the largest robustness radius is determined. Moreover, some alterations of objective functions preserving weak/strict/Henig proper/robust efficiency are studied.

Keywords

Vector optimization Nonsmooth optimization Robust solutions Robustness radius 

Mathematics Subject Classification

90C29 90C31 49J52 

Notes

Acknowledgements

The authors would like to express their gratitude to the editor in chief of JOTA, handling editor, and two anonymous referees 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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Copyright information

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

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Computer Science, College of ScienceUniversity of TehranTehranIran

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