v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 369actually obtained, is independent of binary vector length (12, 13, 20, 24)when averaged over more than 231 problem instances including planar,randomized, and toroidal graphs. 4.43 When failure occurred, large andsmall errors were manifest. That same 98% average accuracy is presumedmaintained when binary vector length is further increased. A Matlabprogram is provided on Wıκımization.4.6.0.0.11 Example. Cardinality/rank problem.d’Aspremont, El Ghaoui, Jordan, & Lanckriet [95] propose approximating apositive semidefinite matrix A∈ S N + by a rank-one matrix having constrainton cardinality c : for 0 < c < Nminimize ‖A − zz T ‖ Fzsubject to cardz ≤ c(828)which, they explain, is a hard problem equivalent tomaximize x T Axxsubject to ‖x‖ = 1card x ≤ c(829)where z √ λx and where optimal solution x ⋆ is a principal eigenvector(1694) (A.5) of A and λ = x ⋆T Ax ⋆ is the principal eigenvalue [159, p.331]when c is true cardinality of that eigenvector. This is principal componentanalysis with a cardinality constraint which controls solution sparsity. Definethe matrix variableX xx T ∈ S N (830)whose desired rank is 1, and whose desired diagonal cardinalitycardδ(X) ≡ cardx (831)4.43 Existence of a polynomial-time approximation to max cut with accuracy provablybetter than 94.11% would refute NP-hardness; which Håstad believes to be highly unlikely.[181, thm.8.2] [182]