v2007.09.13 - Convex Optimization

3.1. CONVEX FUNCTION 189where the norm subscript derives from a binomial coefficient( nk), and‖x‖n n ∆ = ‖x‖ 1‖x‖n1 ∆ = ‖x‖ ∞(437)Finding k largest absolute entries of an n-length vector x is expressible assupremum of 2 k n!/(k!(n − k)!) linear functions of x . [46, exer.6.3(e)]minimize ‖x‖nx∈R n ksubject to x ∈ C≡minimizez∈R n , t∈R , x∈R nsubject tok t + 1 T z−t1 − z ≼ x ≼ t1 + zz ≽ 0x ∈ C(438)3.1.3.2 clippingClipping negative vector entries is accomplished:where, for x = [x i ]∈ R nx + = t ⋆ =‖x + ‖ 1 = minimize 1 T tt∈R nsubject to x ≼ t0 ≼ t[{xi , x i ≥ 00, x i < 0 , i=1... n ](439)(440)(clipping)minimizex∈R n ‖x + ‖ 1subject to x ∈ C≡minimize 1 T tx∈R n , t∈R nsubject to x ≼ t0 ≼ tx ∈ C(441)