Some Remarks on Greenberg–Pierskalla Subdifferentiability of Quasiconvex Functions

Abstract

We observe that a quasiconvex function which is evenly quasiconvex at a point is not necessarily Greenberg–Pierskalla (briefly, G-P) subdifferentiable at that point, but we prove that a quasiconvex function which is upper semicontinuous on the segments of its effective domain is G-P subdifferentiable on the relative interior of this effective domain. We give an application to surrogate duality in quasiconvex programming.

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Acknowledgements

J. E. Martínez-Legaz acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through Grant PGC2018-097960-B-C21 and the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0563). He is affiliated with MOVE (Markets, Organizations and Votes in Economics).

We are grateful to two anonymous referees for their helpful remarks.

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Correspondence to M. Volle.

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Dedicated to Professor Marco A. López on the occasion of his 70th birthday.

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Volle, M., Martínez-Legaz, J.E. Some Remarks on Greenberg–Pierskalla Subdifferentiability of Quasiconvex Functions. Vietnam J. Math. 48, 391–406 (2020). https://doi.org/10.1007/s10013-020-00391-6

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Keywords

  • Quasiconvex functions
  • Greenberg–Pierskalla subdifferential
  • Quasiconvex programming
  • Surrogate duality

Mathematics Subject Classification (2010)

  • 26B25
  • 90C46