Quasiconvex functions: how to separate, if you must!

Frenk, Hans and Dos Santos Gromicho, Joaquim Antonio and Zhang, Shuzhong (2022) Quasiconvex functions: how to separate, if you must! Studia Universitatis Babes-Bolyai Mathematica, 67 (1). pp. 105-128. ISSN 0252-1938 (Print) 2065-961X (Online)

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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.
Item Type: Article
Uncontrolled Keywords: Quasiconvex minimization; Quasidifferentiability; Separation
Divisions: Faculty of Engineering and Natural Sciences
Depositing User: Hans Frenk
Date Deposited: 23 Aug 2022 20:26
Last Modified: 23 Aug 2022 20:26
URI: https://research.sabanciuniv.edu/id/eprint/44077

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