v2010.10.26 - Convex Optimization

382 CHAPTER 4. SEMIDEFINITE PROGRAMMING|x|∫ x ydy−1 |y|+ǫFigure 111: Real absolute value function f 2 (x)= |x| on x∈[−1, 1] fromFigure 68b superimposed upon integral of its derivative at ǫ=0.05 whichsmooths objective function.convex iterationBy convex iteration we mean alternation of solution to (856a) and (862)until convergence. Direction vector y is initialized to 1 until the first fixedpoint is found; which means, the contraction recursion begins calculatinga (1-norm) solution U ⋆ to (854) via problem (856b). Once U ⋆ is found,vector y is updated according to an estimate of discrete image-gradientcardinality c : Sum of the 4n 2 −c smallest entries of |Ψ vec U ⋆ |∈ R 4n2 isthe optimal objective value from a linear program, for 0≤c≤4n 2 − 1 (525)4n ∑2π(|Ψ vec U ⋆ |) i = minimize 〈|Ψ vec U ⋆ | , y〉y∈R 4n2subject to 0 ≼ y ≼ 1i=c+1y T 1 = 4n 2 −c(862)where π is the nonlinear permutation-operator sorting its vector argumentinto nonincreasing order. An optimal solution y to (862), that is anextreme point of its feasible set, is known in closed form: it has 1 in eachentry corresponding to the 4n 2 −c smallest entries of |Ψ vec U ⋆ | and has 0elsewhere (page 336). Updated image U ⋆ is assigned to U t , the contractionis recomputed solving (856b), direction vector y is updated again, andso on until convergence which is guaranteed by virtue of a monotonicallynonincreasing real sequence of objective values in (856a) and (862).