v2010.10.26 - Convex Optimization

4.2. FRAMEWORK 297asminimize ‖ỹ‖ 1ỹsubject to AΛ −1 ỹ = b1 ≽ Λ −1 ỹ ≽ 0(697)where optimal solutiony ⋆ = round(Λ −1 ỹ ⋆ ) (698)The inequality in (697) relaxes Boolean constraint y i ∈ {0, 1} from (680);bounding any solution y ⋆ to a nonnegative unit hypercube whose vertices arebinary numbers. Convex problem (697) is justified by the convex envelopecenv ‖x‖ 0 on {x∈ R n | ‖x‖ ∞ ≤κ} = 1 κ ‖x‖ 1 (1366)Donoho concurs with this particular formulation, equivalently expressible asa linear program via (514).Approximation (697) is therefore equivalent to minimization of an affinefunction (3.2) on a bounded polyhedron, whereas semidefinite programminimize 1 TˆxX∈ S n , ˆx∈R nsubject to A(ˆx + 1) 1[ = b 2] X ˆxG =ˆx T 1δ(X) = 1≽ 0(692)minimizes an affine function on an intersection of the elliptope withhyperplanes. Although the same Boolean solution is obtained fromthis approximation (697) as compared with semidefinite program (692),when given that particular data from Example 4.2.3.1.1, Singer confides acounterexample: Instead, given dataA =[1 01 √20 11√2]then solving approximation (697) yields, b =[11](699)