v2010.10.26 - Convex Optimization

384 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.7 Constraining rank of indefinite matricesExample 4.7.0.0.1, which follows, demonstrates that convex iteration is moregenerally applicable: to indefinite or nonsquare matrices X ∈ R m×n ; not onlyto positive semidefinite matrices. Indeed,findX∈R m×nXsubject to X ∈ CrankX ≤ k≡findX, Y, ZXsubject to X ∈ C[ ] Y XG =X T ZrankG ≤ k(863)Proof. rankG ≤ k ⇒ rankX ≤ k because X is the projection of compositematrix G on subspace R m×n . For symmetric Y and Z , any rank-k positivesemidefinite composite matrix G can be factored into rank-k terms R ;G = R T R where R [B C ] and rankB, rankC ≤ rankR and B ∈ R k×mand C ∈ R k×n . Because Y and Z and X = B T C are variable, (1465)rankX ≤ rankB, rankC ≤ rankR = rankG is tight.So there must exist an optimal direction vector W ⋆ such thatfindX, Y, ZXsubject to X ∈ C[ ] Y XG =X T ZrankG ≤ k≡minimizeX, Y, Z〈G, W ⋆ 〉subject to X ∈ C[ Y XG =X T Z]≽ 0(864)Were W ⋆ = I , by (1688) the optimal resulting trace objective would beequivalent to the minimization of nuclear norm of X over C . This means:(confer p.230) Any nuclear norm minimization problem may have itsvariable replaced with a composite semidefinite variable of the sameoptimal rank but doubly dimensioned.Then Figure 84 becomes an accurate geometrical description of a consequentcomposite semidefinite problem objective. But there are better directionvectors than identity I which occurs only under special conditions: