v2007.09.13 - Convex Optimization

636 APPENDIX E. PROJECTIONE.10.2.1.1 Example. Affine subset ∩ positive semidefinite cone.Consider the problem of finding X ∈ S n that satisfiesX ≽ 0, 〈A j , X〉 = b j , j =1... m (1835)given nonzero A j ∈ S n and real b j . Here we take C 1 to be the positivesemidefinite cone S n + while C 2 is the affine subset of S nC 2 = A = ∆ {X | tr(A j X)=b j , j =1... m} ⊆ S n⎡ ⎤svec(A 1 ) T= {X | ⎣ . ⎦svec X = b}svec(A m ) T∆= {X | A svec X = b}(1836)where b = [b j ] ∈ R m , A ∈ R m×n(n+1)/2 , and symmetric vectorization svec isdefined by (47). Projection of iterate X i ∈ S n on A is: (E.5.0.0.6)P 2 svec X i = svec X i − A † (A svec X i − b) (1837)Euclidean distance from X i to A is thereforedist(X i , A) = ‖X i − P 2 X i ‖ F = ‖A † (A svec X i − b)‖ 2 (1838)Projection of P 2 X i ∆ = ∑ jλ j q j q T j on the positive semidefinite cone (7.1.2) isfound from its eigen decomposition (A.5.2);P 1 P 2 X i =n∑max{0 , λ j }q j qj T (1839)j=1Distance 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 (1840)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.j=1