v2010.10.26 - Convex Optimization

E.10. ALTERNATING PROJECTION 759Distance from P 2 X i to the positive semidefinite cone is thereforen∑dist(P 2 X i , S+) n = ‖P 2 X i − P 1 P 2 X i ‖ F = √ (min{0,λ j }) 2 (2077)When the intersection is empty A ∩ S n +=∅ , the iterates converge to thatpositive semidefinite matrix closest to A in the Euclidean sense. Otherwise,convergence is to some point in the nonempty intersection.Barvinok (2.9.3.0.1) shows that if a solution to (2072) exists, then thereexists an X ∈ A ∩ S n + such that⌊√ ⌋ 8m + 1 − 1rankX ≤2j=1(272)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 matricesare distinct members of the standard orthonormal basis {E lq ∈ S n } (59){ }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} ⊆(2078)are set to constrained entries of variable{ }Xlq√, l = q = 1... nX lq 2 , 1 ≤ l < q ≤ n= {〈X,E lq 〉} (2079)then the equality constraints in (2072) 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 (2074))P 2 svec X i = svec X i − A T (A svec X i − b) (2080)From this we can see that orthogonal projection is achieved simply by settingcorresponding entries of P 2 X i to the known entries of X , while the entriesof P 2 X i remaining are set to corresponding entries of the current iterate X i .