v2007.09.13 - Convex Optimization

3.1. CONVEX FUNCTION 1873.1.3 norm functions, absolute valuewhere |x| = t ⋆ .where ‖x‖ = t ⋆ .‖x‖ 1 = minimize 1 T tt∈R nsubject to −t ≼ x ≼ t‖x‖ 2 = minimizet∈Rwhere max{|x i | , i=1... n} = t ⋆ .subject tot[ ] tI xx T ≽ 0t‖x‖ ∞ = minimize tt∈Rsubject to −t1 ≼ x ≼ t1‖x‖ 1 = minimizeα∈R n , β∈R n 1 T (α + β)subject to α,β ≽ 0x = α − βwhere |x| = α ⋆ + β ⋆ because of complementarity α ⋆T β ⋆ = 0.Optimal solution is norm dependent. [46, p.297] Given set Cminimizex∈R n ‖x‖ 1subject to x ∈ C≡minimize 1 T tx∈R n , t∈R nsubject to −t ≼ x ≼ tx ∈ C(426)(427)(428)(429)(430)minimizex∈R n ‖x‖ 2subject to x ∈ C≡minimizex∈R n , t∈Rsubject tot[ tI xx T tx ∈ C]≽ 0(431)minimize ‖x‖ ∞x∈R nsubject to x ∈ C≡minimize tx∈R n , t∈Rsubject to −t1 ≼ x ≼ t1x ∈ C(432)In R n these norms represent: ‖x‖ 1 is length measured along a grid, ‖x‖ 2 isEuclidean length, ‖x‖ ∞ is maximum |coordinate|.