Chapter 3 Geometry of convex functions - Meboo Publishing ...
Chapter 3 Geometry of convex functions - Meboo Publishing ... Chapter 3 Geometry of convex functions - Meboo Publishing ...
216 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS 3.2 Conic origins of Lagrangian The cone of convex functions, membership relation (476), provides a foundation for what is known as a Lagrangian function. [246, p.398] [274] Consider a conic optimization problem, for proper cone K and affine subset A minimize x f(x) subject to g(x) ∈ K h(x) ∈ A A Cartesian product of convex functions remains convex, so we may write [ fg h ] ⎡ convex w.r.t ⎣ RM + K A (482) ⎤ [ ] ⎡ ⎤ ⎡ ⎤ fg w ⎦ ⇔ [ w T λ T ν T ] convex ∀⎣ λ ⎦∈ G⎣ RM∗ + K ∗ ⎦ h ν A ⊥ (483) where A ⊥ is the normal cone to A . This holds because of equality for h in convexity criterion (475) and because membership relation (429), given point a∈A , becomes h ∈ A ⇔ 〈ν , h − a〉=0 for all ν ∈ G(A ⊥ ) (484) When A = 0, for example, a minimal set of generators G for A ⊥ is a superset of the standard basis for R M (Example E.5.0.0.7) the ambient space of A . A real Lagrangian arises from the scalar term in (483); id est, L [w T λ T ν T ] [ fg h ] = w T f + λ T g + ν T h (485) 3.3 Practical norm functions, absolute value To some mathematicians, “all norms on R n are equivalent” [155, p.53]; meaning, ratios of different norms are bounded above and below by finite predeterminable constants. But to statisticians and engineers, all norms are certainly not created equal; as evidenced by the compressed sensing revolution, begun in 2004, whose focus is predominantly 1-norm. because each summand is considered to be an entry from a vector-valued function.
3.3. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE 217 A norm on R n is a convex function f : R n → R satisfying: for x,y ∈ R n , α∈ R [222, p.59] [155, p.52] 1. f(x) ≥ 0 (f(x) = 0 ⇔ x = 0) (nonnegativity) 2. f(x + y) ≤ f(x) + f(y) (triangle inequality) 3.6 3. f(αx) = |α|f(x) (nonnegative homogeneity) Convexity follows by properties 2 and 3. Most useful are 1-, 2-, and ∞-norm: ‖x‖ 1 = minimize 1 T t t∈R n subject to −t ≼ x ≼ t (486) where |x| = t ⋆ (entrywise absolute value equals optimal t ). 3.7 ‖x‖ 2 = minimize t∈R subject to t [ tI x x T t ] ≽ S n+1 + 0 (487) where ‖x‖ 2 = ‖x‖ √ x T x = t ⋆ . ‖x‖ ∞ = minimize t t∈R subject to −t1 ≼ x ≼ t1 (488) where max{|x i | , i=1... n} = t ⋆ . ‖x‖ 1 = minimize α∈R n , β∈R n 1 T (α + β) subject to α,β ≽ 0 x = α − β (489) where |x| = α ⋆ + β ⋆ because of complementarity α ⋆T β ⋆ = 0 at optimality. (486) (488) (489) represent linear programs, (487) is a semidefinite program. 3.6 with equality iff x = κy where κ ≥ 0. 3.7 Vector 1 may be replaced with any positive [sic] vector to get absolute value, theoretically, although 1 provides the 1-norm.
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3.3. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE 217<br />
A norm on R n is a <strong>convex</strong> function f : R n → R satisfying: for x,y ∈ R n ,<br />
α∈ R [222, p.59] [155, p.52]<br />
1. f(x) ≥ 0 (f(x) = 0 ⇔ x = 0) (nonnegativity)<br />
2. f(x + y) ≤ f(x) + f(y) (triangle inequality) 3.6<br />
3. f(αx) = |α|f(x) (nonnegative homogeneity)<br />
Convexity follows by properties 2 and 3. Most useful are 1-, 2-, and ∞-norm:<br />
‖x‖ 1 = minimize 1 T t<br />
t∈R n<br />
subject to −t ≼ x ≼ t<br />
(486)<br />
where |x| = t ⋆ (entrywise absolute value equals optimal t ). 3.7<br />
‖x‖ 2 = minimize<br />
t∈R<br />
subject to<br />
t<br />
[ tI x<br />
x T t<br />
]<br />
≽<br />
S n+1<br />
+<br />
0<br />
(487)<br />
where ‖x‖ 2 = ‖x‖ √ x T x = t ⋆ .<br />
‖x‖ ∞ = minimize t<br />
t∈R<br />
subject to −t1 ≼ x ≼ t1<br />
(488)<br />
where max{|x i | , i=1... n} = t ⋆ .<br />
‖x‖ 1 = minimize<br />
α∈R n , β∈R n 1 T (α + β)<br />
subject to α,β ≽ 0<br />
x = α − β<br />
(489)<br />
where |x| = α ⋆ + β ⋆ because <strong>of</strong> complementarity α ⋆T β ⋆ = 0 at optimality.<br />
(486) (488) (489) represent linear programs, (487) is a semidefinite program.<br />
3.6 with equality iff x = κy where κ ≥ 0.<br />
3.7 Vector 1 may be replaced with any positive [sic] vector to get absolute value,<br />
theoretically, although 1 provides the 1-norm.
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