v2009.01.01 - Convex Optimization

682 APPENDIX E. PROJECTION
where the convex cone has vertex-description (2.12.2.0.1), for A∈ R n×N
K = {Ay | y ≽ 0} (1912)
and where ‖y‖ ∞ ≤ 1 is the artificial bound. This is a convex optimization
problem having no known closed-form solution, in general. It arises, for
example, in the fitting of hearing aids designed around a programmable
graphic equalizer (a filter bank whose only adjustable parameters are gain
per band each bounded above by unity). [84] The problem is equivalent to a
Schur-form semidefinite program (3.1.7.2)
minimize
y∈R N , t∈R
subject to
t
[
tI x − Ay
(x − Ay) T t
]
≽ 0
(1913)
0 ≼ y ≼ 1
E.9.3
nonexpansivity
E.9.3.0.1 Theorem. Nonexpansivity. [149,2] [92,5.3]
When C ⊂ R n is an arbitrary closed convex set, projector P projecting on C
is nonexpansive in the sense: for any vectors x,y ∈ R n
‖Px − Py‖ ≤ ‖x − y‖ (1914)
with equality when x −Px = y −Py . E.17
⋄
Proof. [47]
‖x − y‖ 2 = ‖Px − Py‖ 2 + ‖(I − P )x − (I − P )y‖ 2
+ 2〈x − Px, Px − Py〉 + 2〈y − Py , Py − Px〉
(1915)
Nonnegativity of the last two terms follows directly from the unique
minimum-distance projection theorem (E.9.1.0.2).
E.17 This condition for equality corrects an error in [69] (where the norm is applied to each
side of the condition given here) easily revealed by counter-example.