v2010.10.26 - Convex Optimization

310 CHAPTER 4. SEMIDEFINITE PROGRAMMINGis known in closed form (p.658): Given ordered diagonalizationG ⋆ = QΛQ T ∈ S N (A.5.1), then direction matrix W = U ⋆ U ⋆T is optimal andextreme where U ⋆ = Q(:, n+1:N)∈ R N×N−n . Eigenvalue vector λ(W) has 1in each entry corresponding to the N − n smallest entries of δ(Λ) and has 0elsewhere. By (221) (223), polar direction −W can be regarded as pointingtoward the set of all rank-n (or less) positive semidefinite matrices whosenullspace contains that of G ⋆ . For that particular closed-form solution W ,consequently (confer (760))N∑λ(G ⋆ ) i = 〈G ⋆ , W 〉 = λ(G ⋆ ) T λ(W) (738)i=n+1This is the connection to cardinality minimization of vectors; 4.25 id est,eigenvalue λ cardinality (rank) is analogous to vector x cardinality via (760).So as not to be misconstrued under closed-form solution W to (1700a):Define (confer (221))S n {(I −W)G(I −W) |G∈ S N } = {X ∈ S N | N(X) ⊇ N(G ⋆ )} (739)as the symmetric subspace of rank≤n matrices whose nullspace containsN(G ⋆ ). Then projection of G ⋆ on S n is (I −W)G ⋆ (I −W). (E.7)Direction of projection is −WG ⋆ W . (Figure 83) tr(WG ⋆ W) is ameasure of proximity to S n because its orthogonal complement isSn ⊥ = {WGW |G∈ S N } ; the point being, convex iteration incorporatingconstrained tr(WGW) = 〈G , W 〉 minimization is not a projection method:certainly, not on these two subspaces.Closed-form solution W to problem (1700a), though efficient, comes witha caveat: there exist cases where this projection matrix solution W does notprovide the most efficient route to an optimal rank-n solution G ⋆ ; id est,direction W is not unique. So we sometimes choose to solve (1700a) insteadof employing a known closed-form solution.4.25 not trace minimization of a nonnegative diagonal matrix δ(x) as in [135,1] [300,2].To make rank-constrained problem (735) resemble cardinality problem (530), we couldmake C an affine subset:find X ∈ S Nsubject to Asvec X = bX ≽ 0rankX ≤ n