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

More documents

Recommendations

Info

248 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS 3.7.2 first-order convexity condition, real function Discretization of w ≽0 in (476) invites refocus to the real-valued function: 3.7.2.0.1 Theorem. Necessary and sufficient convexity condition. [59,3.1.3] [129,I.5.2] [384,1.2.3] [41,1.2] [324,4.2] [300,3] For real differentiable function f(X) : R p×k →R with matrix argument on open convex domain, the condition (conferD.1.7) f(Y ) ≥ f(X) + 〈∇f(X) , Y − X〉 for each and every X,Y ∈ domf (585) is necessary and sufficient for convexity of f . ⋄ When f(X) : R p →R is a real differentiable convex function with vector argument on open convex domain, there is simplification of the first-order condition (585); for each and every X,Y ∈ domf f(Y ) ≥ f(X) + ∇f(X) T (Y − X) (586) From this we can find ∂H − a unique [364, p.220-229] nonvertical [195,B.1.2] hyperplane [ ](2.4), expressed in terms of function gradient, supporting epif X at : videlicet, defining f(Y /∈ domf) ∞ [59,3.1.7] f(X) [ ] Y ∈ epif ⇔ t ≥ f(Y ) ⇒ [ ∇f(X) T −1 ]([ ] [ ]) Y X − ≤ 0 t t f(X) (587) This means, for each and every point X in the domain of a real convex [ function ] f(X) , there exists a hyperplane ∂H − [ in R p ] × R having normal ∇f(X) X supporting the function epigraph at ∈ ∂H −1 f(X) − {[ ] [ ] Y R p [ ∂H − = ∈ ∇f(X) T −1 ]([ ] [ ]) } Y X − = 0 (588) t R t f(X) One such supporting hyperplane (confer Figure 29a) is illustrated in Figure 75 for a convex quadratic. From (586) we deduce, for each and every X,Y ∈ domf ∇f(X) T (Y − X) ≥ 0 ⇒ f(Y ) ≥ f(X) (589)
3.7. GRADIENT 249 f(Y ) [ ∇f(X) −1 ] ∂H − Figure 75: When a real function f is differentiable at each point in its open domain, there is an intuitive geometric interpretation of function convexity in terms of its gradient ∇f and its epigraph: Drawn is a convex quadratic bowl in R 2 ×R (confer Figure 157, p.668); f(Y )= Y T Y : R 2 → R versus Y on some open disc in R 2 . Unique strictly supporting hyperplane ∂H − ∈ R 2 × R (only partially drawn) and its normal vector [ ∇f(X) T −1 ] T at the particular point of support [X T f(X) ] T are illustrated. The interpretation: At each and every coordinate Y , there is a unique hyperplane containing [Y T f(Y ) ] T and supporting the epigraph of convex differentiable f .
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: 3.7. GRADIENT 247 3.7.1.0.3 Example
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 ...

Create successful ePaper yourself

Delete template?

Save as template?