Geoffrion’s proper efficiency in linear fractional vector optimization with unbounded constraint sets

Abstract

Choo (Oper Res 32:216–220, 1984) has proved that any efficient solution of a linear fractional vector optimization problem with a bounded constraint set is properly efficient in the sense of Geoffrion. This paper studies Geoffrion’s properness of the efficient solutions of linear fractional vector optimization problems with unbounded constraint sets. By examples, we show that there exist linear fractional vector optimization problems with the efficient solution set being a proper subset of the unbounded constraint set, which have improperly efficient solutions. Then, we establish verifiable sufficient conditions for an efficient solution of a linear fractional vector optimization to be a Geoffrion properly efficient solution by using the recession cone of the constraint set. For bicriteria problems, it is enough to employ a system of two regularity conditions. If the number of criteria exceeds two, a third regularity condition must be added to the system. The obtained results complement the above-mentioned remarkable theorem of Choo and are analyzed through several interesting examples, including those given by Hoa et al. (J Ind Manag Optim 1:477–486, 2005).

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Acknowledgements

The authors are greatly indebted to the referees for making us familiar with the significant works of Choo  [7], Chew and Choo  [6], Stancu-Minasian  [29], and for valuable suggestions. We thank Prof. Phan Quoc Khanh and Prof. Christiane Tammer for useful discussions on the first draft of this paper.

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Correspondence to N. D. Yen.

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The authors were supported by National Foundation for Science and Technology Development (Vietnam) under Grant No. 101.01-2018.306, Le Quy Don Technical University (Vietnam), Grant MOST 105-2115-M-039-002-MY3 (Taiwan), and the Vietnam Institute for Advanced Study in Mathematics (VIASM, Vietnam).

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Huong, N.T.T., Yao, JC. & Yen, N.D. Geoffrion’s proper efficiency in linear fractional vector optimization with unbounded constraint sets. J Glob Optim 78, 545–562 (2020). https://doi.org/10.1007/s10898-020-00927-7

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Keywords

  • Linear fractional vector optimization
  • Efficient solution
  • Gain-to-loss ratio
  • Geoffrion’s properly efficient solution
  • Unbounded constraint set
  • Direction of recession
  • Regularity condition

Mathematics Subject Classification

  • 90C29
  • 90C32
  • 49K30
  • 49N60