v2009.01.01 - Convex Optimization

600 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS
For B ∈ S N whose eigenvalues λ(B)∈ R N are arranged in nonincreasing
order, and for diagonal matrix Υ∈ S k whose diagonal entries are
arranged in nonincreasing order where 1≤k ≤N , we utilize the
main-diagonal δ operator’s self-adjointness property (1321): [11,4.2]
k∑
i=1
k∑
i=1
Υ ii λ(B) N−i+1 = inf tr(ΥU T BU) = inf
U∈ R N×k
U T U=I
= minimize
V i ∈S N
i=1
U∈ R N×k
U T U=I
δ(Υ) T δ(U T BU)
(
)
∑
tr B k (Υ ii −Υ i+1,i+1 )V i
subject to trV i = i ,
I ≽ V i ≽ 0,
where Υ k+1,k+1 ∆ = 0. We speculate,
k∑
Υ ii λ(B) i = sup
i=1
U∈ R N×k
U T U=I
Alizadeh shows: [10,4.2]
tr(ΥU T BU) = sup
U∈ R N×k
U T U=I
(1582)
i=1... k
i=1... k
δ(Υ) T δ(U T BU) (1583)
Υ ii λ(B) i = minimize
k∑
iµ i + trZ i
µ∈R k , Z i ∈S N i=1
subject to µ i I + Z i − (Υ ii −Υ i+1,i+1 )B ≽ 0, i=1... k
= maximize
V i ∈S N
where Υ k+1,k+1 ∆ = 0.
Z i ≽ 0,
(
)
∑
tr B k (Υ ii −Υ i+1,i+1 )V i
i=1
i=1... k
subject to trV i = i , i=1... k
I ≽ V i ≽ 0, i=1... k (1584)
The largest eigenvalue magnitude µ of A∈ S N
max { |λ(A) i | } = minimize µ
i µ∈R
subject to −µI ≼ A ≼ µI
(1585)