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Cutting Plane Algorithms and Approximate Lower Subdifferentiability

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Abstract

A notion of boundedly ε-lower subdifferentiable functions is introduced and investigated. It is shown that a bounded from below, continuous, quasiconvex function is locally boundedly ε-lower subdifferentiable for every ε>0. Some algorithms of cutting plane type are constructed to solve minimization problems with approximately lower subdifferentiable objective and constraints. In those algorithms an approximate minimizer on a compact set is obtained in a finite number of iterations provided some boundedness assumption be satisfied.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Appliquées, Faculté des sciences, Université de Pau, Av. de l’Université, 64000, Pau, France

    Jean-Paul Penot

  2. Department of Dynamical Systems, Institute of Mathematics, NCSR of Vietnam, P.O. Box 631 Bo Ho, 10000, Hanoi, Vietnam

    Pham Hong Quang

Authors
  1. Jean-Paul Penot
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  2. Pham Hong Quang
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Correspondence to Jean-Paul Penot.

Additional information

Communicated by Jean-Pierre Crouzeix.

The authors are grateful to an anonymous referee for a careful reading and the suggestion of presenting examples illustrating the differences between lower subdifferentiable functions and boundedly approximately lower subdifferentiable functions.

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Penot, JP., Quang, P.H. Cutting Plane Algorithms and Approximate Lower Subdifferentiability. J Optim Theory Appl 148, 455–470 (2011). https://doi.org/10.1007/s10957-010-9762-6

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