v2007.11.26 - Convex Optimization

3.1. CONVEX FUNCTION 1873.1.3 norm functions, absolute valueA vector norm on R n is a function f : R n → R satisfying: for x,y ∈ R n ,α∈ R [110,2.2.1]1. f(x) ≥ 0 (f(x) = 0 ⇔ x = 0) (nonnegativity)2. f(x + y) ≤ f(x) + f(y) (triangle inequality)3. f(αx) = |α|f(x) (nonnegative homogeneity)Most useful of the convex norms are 1-, 2-, and infinity-norm:‖x‖ 1 = minimize 1 T tt∈R nsubject to −t ≼ x ≼ t(427)where |x| = t ⋆ (entrywise absolute value equals optimal t ). 3.5‖x‖ 2 = minimizet∈Rsubject tot[ tI xx T t]≽ 0(428)where ‖x‖ 2 = ‖x‖ ∆ = √ x T x = t ⋆ .‖x‖ ∞ = minimize tt∈Rsubject to −t1 ≼ x ≼ t1(429)where max{|x i | , i=1... n} = t ⋆ .‖x‖ 1 = minimizeα∈R n , β∈R n 1 T (α + β)subject to α,β ≽ 0x = α − β(430)where |x| = α ⋆ + β ⋆ because of complementarity α ⋆T β ⋆ = 0.3.5 Vector 1 may be replaced with any positive [sic] vector to get absolute value,theoretically, although 1 provides the 1-norm and is better from a numerical perspective.