v2009.01.01 - Convex Optimization

598 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTS
the largest eigenvalue, and v N a normalized eigenvector corresponding
to the smallest eigenvalue,
v N = arg inf x T Ax (1574)
‖x‖=1
v 1 = arg sup x T Ax (1575)
‖x‖=1
For A∈ S N having eigenvalues λ(A)∈ R N , consider the unconstrained
nonconvex optimization that is a projection on the rank-1 subset
(2.9.2.1,3.1.8.0.1) of the boundary of positive semidefinite cone S N + :
Defining λ 1 ∆ = max
i
{λ(A) i } and corresponding eigenvector v 1
minimize
x
‖xx T − A‖ 2 F = minimize tr(xx T (x T x) − 2Axx T + A T A)
x
{
‖λ(A)‖ 2 , λ 1 ≤ 0
=
(1576)
‖λ(A)‖ 2 − λ 2 1 , λ 1 > 0
arg minimize
x
‖xx T − A‖ 2 F =
{
0 , λ1 ≤ 0
v 1
√
λ1 , λ 1 > 0
(1577)
Proof. This is simply the Eckart & Young solution from7.1.2:
x ⋆ x ⋆T =
{
0 , λ1 ≤ 0
λ 1 v 1 v T 1 , λ 1 > 0
(1578)
minimize
x
Given nonincreasingly ordered diagonalization A = QΛQ T
Λ = δ(λ(A)) (A.5), then (1576) has minimum value
where
⎧
‖QΛQ T ‖ 2 F = ‖δ(Λ)‖2 , λ 1 ≤ 0
⎪⎨
⎛⎡
‖xx T −A‖ 2 F =
λ 1 Q ⎜⎢
0
⎝⎣
. ..
⎪⎩ ∥
0
⎤ ⎞ ∥ ∥∥∥∥∥∥
2
⎡
⎥ ⎟
⎦− Λ⎠Q T ⎢
=
⎣
∥
F
λ 1
0
.
0
⎤
⎥
⎦− δ(Λ)
∥
2
(1579)
, λ 1 > 0