v2010.10.26 - Convex Optimization

Chapter 3Geometry of convex functionsThe link between convex sets and convex functions is via theepigraph: A function is convex if and only if its epigraph is aconvex set.−Werner FenchelWe limit our treatment of multidimensional functions 3.1 to finite-dimensionalEuclidean space. Then an icon for a one-dimensional (real) convex functionis bowl-shaped (Figure 77), whereas the concave icon is the inverted bowl;respectively characterized by a unique global minimum and maximum whoseexistence is assumed. Because of this simple relationship, usage of the termconvexity is often implicitly inclusive of concavity. Despite iconic imagery,the reader is reminded that the set of all convex, concave, quasiconvex, andquasiconcave functions contains the monotonic functions [203] [215,2.3.5];e.g., [61,3.6, exer.3.46].3.1 vector- or matrix-valued functions including the real functions. Appendix D, with itstables of first- and second-order gradients, is the practical adjunct to this chapter.2001 Jon Dattorro. co&edg version 2010.10.26. All rights reserved.citation: Dattorro, Convex Optimization & Euclidean Distance Geometry,Mεβoo Publishing USA, 2005, v2010.10.26.219