v2007.09.13 - Convex Optimization

188 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS(Ye)minimizex∈R n ‖x‖ 1subject to x ∈ C≡minimize 1 T (α + β)α∈R n , β∈R n , x∈R nsubject to α,β ≽ 0x = α − βx ∈ C(433)All these problems are convex when set C is.3.1.3.1 k smallest/largest entriesSum of the k smallest entries of x∈ R n is the optimal objective value from:for 1≤k ≤nn∑n∑π(x) i = minimize x T yπ(x) i = maximize k t + 1 T zy∈R ni=n−k+1 z∈R n , t∈Rsubject to 0 ≼ y ≼ 1orsubject to x ≽ t1 + z1 T y = kz ≼ 0(434)i=n−k+1which are dual programs, where π(x) 1 = max{x i , i=1... n} where πis the nonlinear permutation operator sorting its vector argument intononincreasing order.Sum of the k largest entries of x∈ R n is the optimal objective value from:[46, exer.5.19]k∑k∑π(x) i = maximize x T yπ(x) i = minimize k t + 1 T zy∈R ni=1z∈R n , t∈Rorsubject to 0 ≼ y ≼ 1subject to x ≼ t1 + z1 T y = kz ≽ 0(435)which are dual programs.Let Πx be a permutation of entries x i such that their absolute valuebecomes arranged in nonincreasing order: |Πx| 1 ≥ |Πx| 2 ≥ · · · ≥ |Πx| n . Byproperties of vector norm, [165, p.59] [109, p.52] sum of the k largest entriesof |x|∈ R n is a norm:i=1‖x‖nk ∆ = k ∑i=1|Πx| i = minimize k t + 1 T zz∈R n , t∈Rsubject to −t1 − z ≼ x ≼ t1 + zz ≽ 0(436)