v2009.01.01 - Convex Optimization

C.2. TRACE, SINGULAR AND EIGEN VALUES 601
is minimized over convex set C by semidefinite program: (confer7.1.5)
id est,
minimize ‖A‖ 2
A
subject to A ∈ C
≡
minimize µ
µ , A
subject to −µI ≼ A ≼ µI
A ∈ C
(1586)
µ ⋆ ∆ = max
i
{ |λ(A ⋆ ) i | , i = 1... N } ∈ R + (1587)
For B ∈ S N whose eigenvalues λ(B)∈ R N are arranged in nonincreasing
order, let Πλ(B) be a permutation of eigenvalues λ(B) such
that their absolute value becomes arranged in nonincreasing
order: |Πλ(B)| 1 ≥ |Πλ(B)| 2 ≥ · · · ≥ |Πλ(B)| N . Then, for 1≤k ≤N
[10,4.3] C.2
k∑
|Πλ(B)| i
i=1
= minimize kµ + trZ
µ∈R , Z∈S N +
subject to µI + Z + B ≽ 0
µI + Z − B ≽ 0
= maximize 〈B , V − W 〉
V ,W ∈ S N +
subject to I ≽ V , W
tr(V + W)=k
(1588)
For diagonal matrix Υ∈ S k whose diagonal entries are arranged in
nonincreasing order where 1≤k ≤N
k∑
Υ ii |Πλ(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
i=1
= maximize
V i ,W i ∈S N
where Υ k+1,k+1 ∆ = 0.
µ i I + Z i − (Υ ii −Υ i+1,i+1 )B ≽ 0, i=1... k
Z i ≽ 0,
(
)
∑
tr B k (Υ ii −Υ i+1,i+1 )(V i − W i )
i=1
i=1... k
subject to tr(V i + W i ) = i , i=1... k
I ≽ V i ≽ 0, i=1... k
I ≽ W i ≽ 0, i=1... k (1589)
C.2 We eliminate a redundant positive semidefinite variable from Alizadeh’s minimization.
There exist typographical errors in [253, (6.49) (6.55)] for this minimization.