v2007.09.17 - Convex Optimization

542 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSFor 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 (1223): [10,4.2]k∑i=1k∑i=1Υ ii λ(B) N−i+1 = inf tr(ΥU T BU) = infU∈ R N×kU T U=I= minimizeV i ∈S Ni=1U∈ R N×kU T U=Iδ(Υ) T δ(U T BU)()∑tr B k (Υ ii −Υ i+1,i+1 )V isubject to trV i = i ,I ≽ V i ≽ 0,where Υ k+1,k+1 ∆ = 0. We speculate,k∑Υ ii λ(B) i = supi=1U∈ R N×kU T U=IAlizadeh shows: [9,4.2]tr(ΥU T BU) = supU∈ R N×kU T U=Ii=1... ki=1... k(1481)δ(Υ) T δ(U T BU) (1482)Υ ii λ(B) i = minimizek∑iµ i + trZ iµ∈R k , Z i ∈S N i=1subject to µ i I + Z i − (Υ ii −Υ i+1,i+1 )B ≽ 0, i=1... k= maximizeV i ∈S Nwhere Υ k+1,k+1 ∆ = 0.Z i ≽ 0,()∑tr B k (Υ ii −Υ i+1,i+1 )V ii=1i=1... ksubject to trV i = i , i=1... kI ≽ V i ≽ 0, i=1... k (1483)The largest eigenvalue magnitude µ of A∈ S Nmax { |λ(A) i | } = minimize µi µ∈Rsubject to −µI ≼ A ≼ µI(1484)