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

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218 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS Over some convex set C given vector constant y or matrix constant Y arg inf x∈C ‖x − y‖ 2 = arg inf x∈C ‖x − y‖2 2 (490) arg inf X∈ C ‖X − Y ‖ F = arg inf X∈ C ‖X − Y ‖2 F (491) are unconstrained convex quadratic problems. Equality does not hold for a sum of norms. (5.4.2.3.2) Optimal solution is norm dependent: [59, p.297] minimize x∈R n ‖x‖ 1 subject to x ∈ C ≡ minimize 1 T t x∈R n , t∈R n subject to −t ≼ x ≼ t x ∈ C (492) minimize x∈R n ‖x‖ 2 subject to x ∈ C ≡ minimize x∈R n , t∈R subject to t [ tI x x T t x ∈ C ] ≽ S n+1 + 0 (493) minimize ‖x‖ ∞ x∈R n subject to x ∈ C ≡ minimize t x∈R n , t∈R subject to −t1 ≼ x ≼ t1 x ∈ C (494) In R n these norms represent: ‖x‖ 1 is length measured along a grid (taxicab distance), ‖x‖ 2 is Euclidean length, ‖x‖ ∞ is maximum |coordinate|. minimize x∈R n ‖x‖ 1 subject to x ∈ C ≡ minimize 1 T (α + β) α∈R n , β∈R n subject to α,β ≽ 0 x = α − β x ∈ C (495) These foregoing problems (486)-(495) are convex whenever set C is. Their equivalence transformations make objectives smooth.
3.3. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE 219 A = {x∈ R 3 |Ax=b} R 3 B 1 = {x∈ R 3 | ‖x‖ 1 ≤ 1} Figure 68: 1-norm ball B 1 is convex hull of all cardinality-1 vectors of unit norm (its vertices). Ball boundary contains all points equidistant from origin in 1-norm. Cartesian axes drawn for reference. Plane A is overhead (drawn truncated). If 1-norm ball is expanded until it kisses A (intersects ball only at boundary), then distance (in 1-norm) from origin to A is achieved. Euclidean ball would be spherical in this dimension. Only were A parallel to two axes could there be a minimum cardinality least Euclidean norm solution. Yet 1-norm ball offers infinitely many, but not all, A-orientations resulting in a minimum cardinality solution. (1-norm ball is an octahedron in this dimension while ∞-norm ball is a cube.)
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: 3.3. PRACTICAL NORM FUNCTIONS, ABSO
Page 11 and 12: 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 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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