v2009.01.01 - Convex Optimization

C.2. TRACE, SINGULAR AND EIGEN VALUES 597
For X ∈ S m , Y ∈ S n , A∈ C ⊆ R m×n for set C convex, and σ(A)
denoting the singular values of A [113,3]
minimize
A
∑
σ(A) i
subject to A ∈ C
i
≡
1
minimize 2
A , X , Y
subject to
tr X + trY
[ ] X A
A T ≽ 0
Y
A ∈ C
(1569)
For A∈ S N + and β ∈ R
β trA = maximize tr(XA)
X∈ S N
subject to X ≼ βI
(1570)
But the following statement is numerically stable, preventing an
unbounded solution in direction of a 0 eigenvalue:
maximize sgn(β) tr(XA)
X∈ S N
subject to X ≼ |β|I
X ≽ −|β|I
(1571)
where β trA = tr(X ⋆ A). If β ≥ 0 , then X ≽−|β|I ← X ≽ 0.
For A∈ S N having eigenvalues λ(A)∈ R N , its smallest and largest
eigenvalue is respectively [10,4.1] [38,I.6.15] [176,4.2] [206,2.1]
[207]
min{λ(A) i } = inf
i
‖x‖=1 xT Ax = minimize
X∈ S N +
subject to trX = 1
max{λ(A) i } = sup x T Ax = maximize
i
‖x‖=1
X∈ S N +
subject to trX = 1
tr(XA) = maximize t
t∈R
subject to A ≽ tI
(1572)
tr(XA) = minimize t
t∈R
subject to A ≼ tI
(1573)
The smallest eigenvalue of any symmetric matrix is always a concave
function of its entries, while the largest eigenvalue is always convex.
[53, exmp.3.10] For v 1 a normalized eigenvector of A corresponding to