Quasiconvex functions: how to separate, if you must!

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 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 {\em 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.