Quasiconvex functions: how to separate, if you must!

Johannes Bartholomeus Gerardus Frenk, Joaquim Antonio dos Santos Gromicho, Shuzhong Zhang

Research output: Contribution to journalArticlepeer-review


Since quasiconvex functions have convex lower level sets it is possible to minimize them by means of separating hyperplanes. An example of such a procedure, well-known for convex functions, is the subgradient method. However, to find the normal vector of a separating hyperplane is in general not easy for the quasiconvex case. This paper attempts to gain some insight into the computational aspects of determining such a normal vector and the geometry of lower level sets of quasiconvex functions. In order to do so, the directional differentiability of quasiconvex functions is thoroughly studied. As a consequence of that study, it is shown that an important subset of quasiconvex functions belongs to the class of quasidifferentiable functions. The main emphasis is, however, on computing actual separators. Some important examples are worked out for illustration.

Original languageEnglish (US)
Pages (from-to)105-128
Number of pages24
JournalStudia Universitatis Babes-Bolyai Mathematica
Issue number1
StatePublished - 2022
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2022, Studia Universitatis Babes-Bolyai Mathematica. All rights reserved.


  • Quasiconvex minimization
  • Quasidifferentiability
  • Separation


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