v2010.10.26 - Convex Optimization

660 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSFor B ∈ S N whose eigenvalues λ(B)∈ R N are arranged in nonincreasingorder, let Πλ(B) be a permutation of eigenvalues λ(B) suchthat their absolute value becomes arranged in nonincreasingorder: |Πλ(B)| 1 ≥ |Πλ(B)| 2 ≥ · · · ≥ |Πλ(B)| N . Then, for 1≤k ≤N[11,4.3] C.3k∑|Πλ(B)| ii=1k∑i=1= minimize kµ + trZµ∈R , Z∈S N +subject to µI + Z + B ≽ 0µI + Z − B ≽ 0= maximize 〈B , V − W 〉V ,W ∈ S N +subject to I ≽ V , Wtr(V + W)=k(1707)For diagonal matrix Υ∈ S k whose diagonal entries are arranged innonincreasing order where 1≤k ≤Nk∑Υ ii |Πλ(B)| i = minimize 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 ,W i ∈S Nwhere Υ k+1,k+1 0.µ i I + Z i − (Υ ii −Υ i+1,i+1 )B ≽ 0, i=1... kZ i ≽ 0,()∑tr B k (Υ ii −Υ i+1,i+1 )(V i − W i )i=1i=1... ksubject to tr(V i + W i ) = i , i=1... kI ≽ V i ≽ 0, i=1... kI ≽ W i ≽ 0, i=1... k (1708)C.2.0.0.3 Exercise. Weighted sum of largest eigenvalues.Prove (1702).C.3 We eliminate a redundant positive semidefinite variable from Alizadeh’s minimization.There exist typographical errors in [292, (6.49) (6.55)] for this minimization.