v2009.01.01 - Convex Optimization

698 APPENDIX E. PROJECTION
Barvinok (2.9.3.0.1) shows that if a point feasible with (1948) exists,
then there exists an X ∈ A ∩ S n + such that
⌊√ ⌋ 8m + 1 − 1
rankX ≤
(245)
2
E.10.2.1.2 Example. Semidefinite matrix completion.
Continuing Example E.10.2.1.1: When m≤n(n + 1)/2 and the A j matrices
are distinct members of the standard orthonormal basis {E lq ∈ S n } (52)
{ }
el e T
{A j ∈ S n l , l = q = 1... n
, j =1... m} ⊆ {E lq } =
√1
2
(e l e T q + e q e T l ), 1 ≤ l < q ≤ n
and when the constants b j
X = ∆ [X lq ]∈ S n
{b j , j =1... m} ⊆
(1954)
are set to constrained entries of variable
{ }
Xlq
√
, l = q = 1... n
X lq 2 , 1 ≤ l < q ≤ n
= {〈X,E lq 〉} (1955)
then the equality constraints in (1948) fix individual entries of X ∈ S n . Thus
the feasibility problem becomes a positive semidefinite matrix completion
problem. Projection of iterate X i ∈ S n on A simplifies to (confer (1950))
P 2 svec X i = svec X i − A T (A svec X i − b) (1956)
From this we can see that orthogonal projection is achieved simply by
setting corresponding entries of P 2 X i to the known entries of X , while
the remaining entries of P 2 X i are set to corresponding entries of the current
iterate X i .
Using this technique, we find a positive semidefinite completion for
⎡ ⎤
4 3 ? 2
⎢ 3 4 3 ?
⎥
⎣ ? 3 4 3 ⎦ (1957)
2 ? 3 4
Initializing the unknown entries to 0, they all converge geometrically to
1.5858 (rounded) after about 42 iterations.