v2010.10.26 - Convex Optimization

272 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.log-convex ⇒ convex ⇒ quasiconvex.concave ⇒ quasiconcave ⇐ log-concave ⇐ positive concave. 3.294. The line theorem (3.7.3.0.1) translates identically to quasiconvexity(quasiconcavity). [61,3.4.2]5.g convex ⇔ −g concaveg quasiconvex ⇔ −g quasiconcaveg log-convex ⇔ 1/g log-concave(translation, homogeneity) Function convexity, concavity,quasiconvexity, and quasiconcavity are invariant to offset andnonnegative scaling.6. (affine transformation of argument) Composition g(h(X)) of convex(concave) function g with any affine function h : R m×n → R p×kremains convex (concave) in X ∈ R m×n , where h(R m×n ) ∩ domg ≠ ∅ .[199,B.2.1] Likewise for the quasiconvex (quasiconcave) functions g .7. – Nonnegatively weighted sum of convex (concave) functionsremains convex (concave). (3.1.2.1.1)– Nonnegatively weighted nonzero sum of strictly convex(concave) functions remains strictly convex (concave).– Nonnegatively weighted maximum (minimum) of convex 3.30(concave) functions remains convex (concave).– Pointwise supremum (infimum) of convex (concave) functionsremains convex (concave). (Figure 74) [307,5] –Sum of quasiconvex functions not necessarily quasiconvex.– Nonnegatively weighted maximum (minimum) of quasiconvex(quasiconcave) functions remains quasiconvex (quasiconcave).– Pointwise supremum (infimum) of quasiconvex (quasiconcave)functions remains quasiconvex (quasiconcave).3.29 Log-convex means: logarithm of function f is convex on dom f .3.30 Supremum and maximum of convex functions are proven convex by intersection ofepigraphs.