v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
194 CHAPTER 2. CONVEX GEOMETRY
Chapter 3 Geometry of convex functions The link between convex sets and convex functions is via the epigraph: A function is convex if and only if its epigraph is a convex set. −Stephen Boyd & Lieven Vandenberghe [53,3.1.7] We limit our treatment of multidimensional functions 3.1 to finite-dimensional Euclidean space. Then an icon for a one-dimensional (real) convex function is bowl-shaped (Figure 67), whereas the concave icon is the inverted bowl; respectively characterized by a unique global minimum and maximum whose existence is assumed. Because of this simple relationship, usage of the term convexity is often implicitly inclusive of concavity in the literature. Despite the iconic imagery, the reader is reminded that the set of all convex, concave, quasiconvex, and quasiconcave functions contains the monotonic functions [177] [186,2.3.5]; e.g., [53,3.6, exer.3.46]. 3.1 vector- or matrix-valued functions including the real functions. Appendix D, with its tables of first- and second-order gradients, is the practical adjunct to this chapter. 2001 Jon Dattorro. CO&EDG version 2009.01.01. All rights reserved. Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry, Meboo Publishing USA, 2005. 195
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194 CHAPTER 2. CONVEX GEOMETRY
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