v2010.10.26 - Convex Optimization

234 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSwhere gradient of k-largest norm is an optimal solution to a convex problem:⎫‖x‖n = maximize y T xk y∈R nsubject to 0 ≼ y ≼ 1y T 1 = k ⎪⎬, x ≽ 0 (533)∇‖x‖n = arg maximize y T xk y∈R nsubject to 0 ≼ y ≼ 1 ⎪⎭y T 1 = k3.2.2.1.3 Exercise. k-largest norm gradient.Prove (532). Find ∇‖x‖ 1 and ∇‖x‖nkon R n . 3.123.2.3 clippingZeroing negative vector entries under 1-norm is accomplished:‖x + ‖ 1 = minimize 1 T tt∈R nsubject to x ≼ t0 ≼ t(534)where, for x=[x i , i=1... n]∈ R nx + t ⋆ =(clipping)[xi , x i ≥ 00, x i < 0}, i=1... n]= 1 (x + |x|) (535)2minimizex∈R n ‖x + ‖ 1subject to x ∈ C≡minimize 1 T tx∈R n , t∈R nsubject to x ≼ t0 ≼ tx ∈ C(536)3.12 Hint:D.2.1.