v2009.01.01 - Convex Optimization

3.1. CONVEX FUNCTION 199
3.1.3 norm functions, absolute value
A vector norm on R n is a function f : R n → R satisfying: for x,y ∈ R n ,
α∈ R [134,2.2.1]
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 t
t∈R n
subject to −t ≼ x ≼ t
(450)
where |x| = t ⋆ (entrywise absolute value equals optimal t ). 3.5
‖x‖ 2 = minimize
t∈R
subject to
t
[ tI x
x T t
]
≽ 0
(451)
where ‖x‖ 2 = ‖x‖ ∆ = √ x T x = t ⋆ .
‖x‖ ∞ = minimize t
t∈R
subject to −t1 ≼ x ≼ t1
(452)
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
(453)
where |x| = α ⋆ + β ⋆ because of complementarity α ⋆T β ⋆ = 0 at optimality.
(450) (452) (453) represent linear programs, (451) is a semidefinite program.
3.5 Vector 1 may be replaced with any positive [sic] vector to get absolute value,
theoretically, although 1 provides the 1-norm.