v2007.09.13 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 2854.4.3.0.8 Example. Cardinality/sparsity problem.d’Aspremont et alii [65] propose approximating a positive semidefinite matrixA ∈ S N + by a rank-one matrix having a constraint on cardinality c : for0 < c < Nminimize ‖A − zz T ‖ Fz(685)subject to cardz ≤ cwhich, they explain, is a hard problem equivalent tomaximize x T Axxsubject to ‖x‖ = 1cardx ≤ c(686)where z ∆ = √ λ x and where optimal solution x ⋆ is a principal eigenvector(1469) (A.5) of A and λ = x ⋆T Ax ⋆ is the principal eigenvalue when c istrue cardinality of that eigenvector. This is principal component analysiswith a cardinality constraint which controls solution sparsity. Define thematrix variableX ∆ = xx T ∈ S N (687)whose desired rank is 1, and whose desired diagonal cardinalitycardδ(X) ≡ cardx (688)is equivalent to cardinality c of vector x . Then we can transform cardinalityproblem (686) to an equivalent problem in new variable X : 4.32maximizeX∈S N 〈X , A〉subject to 〈X , I〉 = 1(X ≽ 0)rankX = 1cardδ(X) ≤ c(689)We transform problem (689) to an equivalent convex problem byintroducing two direction matrices: W to achieve desired cardinality4.32 A semidefiniteness constraint X ≽ 0 is not required, theoretically, because positivesemidefiniteness of a rank-1 matrix is enforced by symmetry. (Theorem A.3.1.0.7)