Quasiconvex functions: how to separate, if you must!-Reference-Cited by-同舟云学术

Quasiconvex functions: how to separate, if you must!

Published:2022-03-10 Issue:1 Volume:67 Page:105-128
ISSN:0252-1938
Container-title:Studia Universitatis Babes-Bolyai Matematica
language:
Short-container-title:Stud. Univ. Babes-Bolyai Math.

Author:

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

Abstract

"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 nd 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 di erentiability 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."

Publisher

Babes-Bolyai University

Subject

General Mathematics

Reference22 articles.

1. "[1] Clarke, F.H., Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983.

2. [2] Crouzeix, J.-P., Some di erentiability properties of quasiconvex functions on Rn, In: Optimization and Optimal Control (Proc. Conf., Math. Res. Inst., Oberwolfach, 1980), vol. 30 of Lecture Notes in Control and Information Sci., Springer, Berlin - New York, 1981, pp. 9-20.

3. [3] Crouzeix, J.-P., About differentiability of order one of quasiconvex functions on Rn, J. Optim. Theory Appl., 36(1982), no. 3, 367-385.

4. [4] Crouzeix, J.-P., A review of continuity and di erentiability properties of quasiconvex functions on Rn, In: Convex Analysis and Optimization (London, 1980), vol. 57 of Res. Notes in Math., Pitman, Boston, Mass.-London, 1982, pp. 18-34.

5. [5] Dem'yanov, V.F., Dixon, L.C.W., Quasidi erential Calculus, vol. 29, North-Holland Sole Distributor for the U.S.A., Elsevier Science Publishers, Amsterdam - New York, N.Y., USA, 1986.