v2007.09.17 - Convex Optimization

E.10. ALTERNATING PROJECTION 639Barvinok (2.9.3.0.1) shows that if a point feasible with (1840) exists,then there exists an X ∈ A ∩ S n + such that⌊√ ⌋ 8m + 1 − 1rankX ≤(232)2E.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 matricesare distinct members of the standard orthonormal basis {E lq ∈ S n } (50){ }el e T{A j ∈ S n l , l = q = 1... n, j =1... m} ⊆ {E lq } =√12(e l e T q + e q e T l ), 1 ≤ l < q ≤ nand when the constants b jX = ∆ [X lq ]∈ S n{b j , j =1... m} ⊆(1846)are set to constrained entries of variable{ }Xlq√, l = q = 1... nX lq 2 , 1 ≤ l < q ≤ n= {〈X,E lq 〉} (1847)then the equality constraints in (1840) fix individual entries of X ∈ S n . Thusthe feasibility problem becomes a positive semidefinite matrix completionproblem. Projection of iterate X i ∈ S n on A simplifies to (confer (1842))P 2 svec X i = svec X i − A T (A svec X i − b) (1848)From this we can see that orthogonal projection is achieved simply bysetting corresponding entries of P 2 X i to the known entries of X , whilethe remaining entries of P 2 X i are set to corresponding entries of the currentiterate X i .Using this technique, we find a positive semidefinite completion for⎡ ⎤4 3 ? 2⎢ 3 4 3 ?⎥⎣ ? 3 4 3 ⎦ (1849)2 ? 3 4Initializing the unknown entries to 0, they all converge geometrically to1.5858 (rounded) after about 42 iterations.