Optimization of Globally Convex Functions

Abstract

Convex functions have nice properties with respect to both minimization and maximization. Similar properties are established here for functions that are permitted to have bad local behavior but are globally convex in the sense that they behave “convexly” on triples of collinear points that are widely dispersed. The results illustrate a development that seems desirable in the interest of more realistic mathematical modeling: the “globalization” of important function properties. In connection with the maximization of globally convex functions over convex bodies in a given finite-dimensional normed space E, there is interest in estimating the maximum, for points c of bodies $C \subset E$, of the ratio between two measures of how close c comes to being an extreme point of C. Good estimates are obtained for the cases in which E is Euclidean or has the “max” norm.