Chapter 3 Geometry of convex functions - Meboo Publishing ...

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262 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS 3.10 Salient properties of convex and quasiconvex functions 1. A convex function is assumed continuous but not necessarily differentiable on the relative interior of its domain. [301,10] A quasiconvex function is not necessarily a continuous function. 2. convex epigraph ⇔ convexity ⇒ quasiconvexity ⇔ convex sublevel sets. convex hypograph ⇔ concavity ⇒ quasiconcavity ⇔ convex superlevel. monotonicity ⇒ quasilinearity ⇔ convex level sets. 3. log-convex ⇒ convex ⇒ quasiconvex. concave ⇒ quasiconcave ⇐ log-concave ⇐ positive concave. 3.26 4. (homogeneity) Convexity, concavity, quasiconvexity, and quasiconcavity are invariant to nonnegative scaling of function. g convex ⇔ −g concave g quasiconvex ⇔ −g quasiconcave g log-convex ⇔ 1/g log-concave 5. The line theorem (3.8.3.0.1) translates identically to quasiconvexity (quasiconcavity). [59,3.4.2] 6. (affine transformation of argument) Composition g(h(X)) of convex (concave) function g with any affine function h : R m×n → R p×k remains convex (concave) in X ∈ R m×n , where h(R m×n ) ∩ domg ≠ ∅ . [195,B.2.1] Likewise for the quasiconvex (quasiconcave) functions g . 3.26 Log-convex means: logarithm of function f is convex on dom f .
3.10. SALIENT PROPERTIES 263 7. – Nonnegatively weighted sum of convex (concave) functions remains 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 3.27 (concave) functions remains convex (concave). – Pointwise supremum (infimum) of convex (concave) functions remains convex (concave). (Figure 72) [301,5] – 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). 3.27 Supremum and maximum of convex functions are proven convex by intersection of epigraphs.
Page 1 and 2: Chapter 3 Geometry of convex functi
Page 3 and 4: 3.1. CONVEX FUNCTION 213 f 1 (x) f
Page 5 and 6: 3.1. CONVEX FUNCTION 215 Rf (b) f(X
Page 7 and 8: 3.3. PRACTICAL NORM FUNCTIONS, ABSO
Page 17 and 18: 3.4. INVERTED FUNCTIONS AND ROOTS 2
Page 19 and 20: 3.5. AFFINE FUNCTION 229 3.4.1.3 po
Page 21 and 22: 3.5. AFFINE FUNCTION 231 3.5.0.0.2
Page 23 and 24: 3.6. EPIGRAPH, SUBLEVEL SET 233 q(x
Page 25 and 26: 3.6. EPIGRAPH, SUBLEVEL SET 235 To
Page 27 and 28: 3.6. EPIGRAPH, SUBLEVEL SET 237 con
Page 29 and 30: 3.6. EPIGRAPH, SUBLEVEL SET 239 3.6
Page 31 and 32: 3.7. GRADIENT 241 2 1.5 1 0.5 Y 2 0
Page 33 and 34: 3.7. GRADIENT 243 From (1749) andD.
Page 35 and 36: 3.7. GRADIENT 245 This equivalence
Page 37 and 38: 3.7. GRADIENT 247 3.7.1.0.3 Example
Page 39 and 40: 3.7. GRADIENT 249 f(Y ) [ ∇f(X)
Page 41 and 42: 3.7. GRADIENT 251 meaning, the grad
Page 43 and 44: 3.8. MATRIX-VALUED CONVEX FUNCTION
Page 49 and 50: 3.9. QUASICONVEX 259 3.8.3.0.6 Exam
Page 51: 3.9. QUASICONVEX 261 3.9.0.0.2 Defi

Chapter 3 Geometry of convex functions - Meboo Publishing ...

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