Quasiconvex Programming

Abstract

This chapter starts by introducing in Section 5.1 a so-called quasiconvex program. In order to solve such a program by the ellipsoid method it should be computationally possible to construct a hyperplane separating a given point from its corresponding strict lower level set. In spite of the convexity of this set such a “computable” separation is not a trivial matter for a quasiconvex function. Therefore, Section 5.2, based on Ref. [39], is devoted to this topic. This section starts by analyzing the cone of descent directions of a quasiconvex function at a given point and using the properties of this cone the local geometry of the corresponding strict lower level set is studied in Subsection 5.2.1. In the case where the cone of descent directions also contains strict descent directions it turns out that the local information given by this smaller cone completely determines all hyperplanes separating the considered point from its strict lower level set. This observation enables us in Subsection 5.2.2 to show that implementing the separation oracle can be done by solving an optimization problem and so it is in principle possible to construct separating hyperplanes if the cone of strict descent directions is nonempty. Moreover, this optimization problem also determines whether the cone of strict descent directions is empty or not. Since solving the optimization problem may be a difficult computational task we discuss in Section 5.3 instances of quasiconvex functions for which this optimization problem can be replaced by an easier membership problem. Since in the absence of strict descent directions the local information given by the cone of descent directions is not sufficient to construct an implementable separation oracle we explain in Section 5.4 what to do in such a situation and the convergence of the corresponding modified ellipsoid algorithm is proven in Section 5.5.