v2010.10.26 - Convex Optimization

658 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS(Ky Fan, 1949) For B ∈ S N whose eigenvalues λ(B)∈ R N are arrangedin nonincreasing order, and for 1≤k ≤N [11,4.1] [215] [202,4.3.18][359,2] [239,2.1] [240]N∑i=N−k+1λ(B) i = infU∈ R N×kU T U=Itr(UU T B) = minimizeX∈ S N +tr(XB)subject to X ≼ ItrX = k(a)k∑λ(B) i = supi=1U∈ R N×kU T U=I= maximize (k − N)µ + tr(B − Z)µ∈R , Z∈S N +subject to µI + Z ≽ Btr(UU T B) = maximizeX∈ S N +tr(XB)subject to X ≼ ItrX = k(b)(c)= minimize kµ + trZµ∈R , Z∈S N +subject to µI + Z ≽ B(d)(1700)Given ordered diagonalization B = QΛQ T , (A.5.1) then an optimalU for the infimum is U ⋆ = Q(:, N − k+1:N)∈ R N×k whereasU ⋆ = Q(:, 1:k)∈ R N×k for the supremum is more reliably computed.In both cases, X ⋆ = U ⋆ U ⋆T . Optimization (a) searches the convex hullof outer product UU T of all N ×k orthonormal matrices. (2.3.2.0.1)For B ∈ S N whose eigenvalues λ(B)∈ R N are arranged in nonincreasingorder, and for diagonal matrix Υ∈ S k whose diagonal entries arearranged in nonincreasing order where 1≤k ≤N , we utilize themain-diagonal δ operator’s self-adjointness property (1418): [12,4.2]k∑Υ ii λ(B) N−i+1 = inf tr(ΥU T BU) = inf δ(Υ) T δ(U T BU)i=1U∈ R N×kU∈ R N×kU T U=I= minimizeV i ∈S Ni=1U T U=I()∑tr B k (Υ ii −Υ i+1,i+1 )V isubject to trV i = i ,I ≽ V i ≽ 0,(1701)i=1... ki=1... k