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

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228 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS 3.4.1.1 positive Then we have the equivalent semidefinite program for maximizing a concave function x α , for quantized 0≤α 0 x ∈ C ≡ maximize x∈R , y∈R q+1 subject to y q [ yi−1 y i y i x b i ] ≽ 0 , i=1... q x ∈ C (522) where nonnegativity <strong>of</strong> y q is enforced by maximization; id est, x > 0, y q ≤ x α ⇔ [ yi−1 y i y i x b i ] ≽ 0 , i=1... q (523) 3.4.1.2 negative It is also desirable to implement an objective <strong>of</strong> the form x −α for positive α . The technique is nearly the same as before: for quantized 0≤α 0 rather x ∈ C x > 0, z ≥ x −α ≡ ⇔ minimize x , z∈R , y∈R q+1 subject to [ yi−1 y i y i x b i z [ yi−1 y i y i x b i ] [ ] z 1 ≽ 0 1 y q ≽ 0 , i=1... q x ∈ C (524) ] [ ] z 1 ≽ 0 1 y q ≽ 0 , i=1... q (525)
3.5. AFFINE FUNCTION 229 3.4.1.3 positive inverted Now define vector t=[t i , i=0... q] with t 0 =1. To implement an objective x 1/α for quantized 0≤α
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: 3.4. INVERTED FUNCTIONS AND ROOTS 2
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 and 52: 3.9. QUASICONVEX 261 3.9.0.0.2 Defi
Page 53: 3.10. SALIENT PROPERTIES 263 7. - N

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

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