v2007.11.26 - Convex Optimization

4.2. FRAMEWORK 2454.2.3.1.2 Example. Optimization on elliptope versus 1-norm polyhedronfor minimum cardinality Boolean Example 4.2.3.1.1. A minimumcardinality problem is typically formulated via, what is by now, the standardpractice [82] of column normalization applied to a 1-norm problem surrogatelike (582). Suppose we define a diagonal matrix⎡⎤‖A(:,1)‖ 2 0Λ =∆ ‖A(:, 2)‖ 2 ⎢⎥⎣... ⎦ ∈ S6 (593)0 ‖A(:, 6)‖ 2used to normalize the columns (assumed nonzero) of given noiseless datamatrix A . Then approximate the minimum cardinality Boolean problemaswhere optimal solutionminimize ‖x‖ 0xsubject to Ax = bx i ∈ {0, 1} ,i=1... nminimize ‖ỹ‖ 1ỹsubject to AΛ −1 ỹ = b1 ≽ Λ −1 ỹ ≽ 0(577)(594)y ⋆ = round(Λ −1 ỹ ⋆ ) (595)The inequality in (594) relaxes Boolean constraint y i ∈ {0, 1} from (577);serving to bound any solution y ⋆ to a unit cube whose vertices are binarynumbers. Convex problem (594) is justified by the convex envelopecenv ‖x‖ 0 on {x∈ R n | ‖x‖ ∞ ≤κ} = 1 κ ‖x‖ 1 (1197)Donoho concurs with this particular formulation equivalently expressible asa linear program via (431).Approximation (594) is therefore equivalent to minimization of an affinefunction on a bounded polyhedron, whereas semidefinite programminimize 1 T ˆxX∈ S n , ˆx∈R nsubject to A(ˆx + 1) 1[ = b 2] X ˆxG =ˆx T 1δ(X) = 1≽ 0(589)