Minimization of quasi-convex symmetric and of discretely quasi-convex symmetric functions

Abstract

The main results, given in Section 3, deal with the effective computation of a minimum point x ∗ of a discretely quasi-convex symmetric function f In particular, an explicit formula for x ∗ is given if f is to be minimized under a linear symmetric constraint. Most important is the case where fis the restriction of a quasi-convex symmetric function [ftilde] on a convex symmetric subset of ℝn to a discretely convex symmetric subset of ℤn. It is shown how in this case x ∗ can be found by computing f(x) at only n points x, determined by a minimum point of [ftilde]. The latter can be found easily by minimizing a quasi-convex function of a single real variable, determined by [ftilde] This point is elaborated in Section 2. The methods are illustrated by several examples. Finally we point out that most results hold also for (discretely) Schurconvex functions