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Optimality Conditions and Constraint Qualifications for Quasiconvex Programming

  • Satoshi SuzukiEmail author
Regular Paper
  • 29 Downloads

Abstract

In mathematical programming, various kinds of optimality conditions have been introduced. In the research of optimality conditions, some types of subdifferentials play an important role. Recently, by using Greenberg–Pierskalla subdifferential and Martínez-Legaz subdifferential, necessary and sufficient optimality conditions for quasiconvex programming have been introduced. On the other hand, constraint qualifications are essential elements for duality theory in mathematical programming. Over the last decade, necessary and sufficient constraint qualifications for duality theorems have been investigated extensively. Recently, by using the notion of generator, necessary and sufficient constraint qualifications for Lagrange-type duality theorems have been investigated. However, constraint qualifications for optimality conditions in terms of Greenberg–Pierskalla subdifferential and Martínez-Legaz subdifferential have not been investigated yet. In this paper, we study optimality conditions and constraint qualifications for quasiconvex programming. We introduce necessary and sufficient optimality conditions in terms of Greenberg–Pierskalla subdifferential, Martínez-Legaz subdifferential and generators. We investigate necessary and/or sufficient constraint qualifications for these optimality conditions. Additionally, we show some equivalence relations between duality results for convex and quasiconvex programming.

Keywords

Quasiconvex programming Optimality condition Constraint qualification Generator of a quasiconvex function 

Mathematics Subject Classification

90C26 90C46 49J52 

Notes

Acknowledgements

The author is grateful to anonymous referees for many comments and suggestions which improved the quality of the paper.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesShimane UniversityShimaneJapan

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