Approximate Proper Solutions in Vector Equilibrium Problems

Limit Behavior and Linear Scalarization Results

Abstract

The paper concerns with properties of approximate Benson and Henig proper efficient solutions of vector equilibrium problems. Relationships between both kinds of solutions and the approximate weak efficient solutions of the problem are stated. Roughly speaking, it is proved that they coincide provided that the ordering cone of the problem can be separated from another closed cone by an approximating sequence of dilating cones. As a result, the limit behavior of approximate Benson and Henig proper efficient solutions when the error tends to zero is studied. It is shown that efficient and proper efficient solutions of the problem can be obtained as the limit of approximate Benson and Henig proper efficient solutions whenever a suitable domination set is considered. Finally, optimality conditions for approximate Benson and Henig proper efficient solutions are derived by approximate solutions of scalar equilibrium problems. They are formulated via linear scalarization and the necessary conditions hold true in problems satisfying a kind of nearly subconvexlikeness condition. The main results of the paper generalize some recent ones because they involve weaker assumptions.

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Acknowledgements

The author is very grateful to the referees for their valuable comments and suggestions. This research was partially supported by Ministerio de Economía y Competitividad (Spain) under project MTM2015-68103-P (MINECO/FEDER), by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) (Spain) and Fondo Europeo de Desarrollo Regional (FEDER) under project PGC2018-096899-B-I00 (MCIU/AEI/FEDER, UE).

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Correspondence to C. Gutiérrez.

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This work is dedicated to Professor Marco Antonio López Cerdá on the occasion of his 70th birthday.

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Gutiérrez, C. Approximate Proper Solutions in Vector Equilibrium Problems. Vietnam J. Math. (2020). https://doi.org/10.1007/s10013-020-00416-0

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Keywords

  • Vector equilibrium problem
  • Proper efficient solution
  • Approximate solution
  • Dilating cone
  • Approximating sequence of cones
  • Generalized convexity
  • Linear scalarization

Mathematics Subject Classification (2010)

  • 90C33
  • 49J40
  • 90C29
  • 90C48