v2010.10.26 - Convex Optimization

758 APPENDIX E. PROJECTIONx ki represents unique minimum-distance projection of x k+1,i on convex set kat iteration i . So a good convergence measure is total monotonic sequenceε i L∑‖x ki − x k,i+1 ‖ , i=0, 1, 2... (2071)k=1where limi→∞ε i = 0 whether or not the intersection is nonempty.E.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 (2072)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}(2073)where b = [b j ] ∈ R m , A ∈ R m×n(n+1)/2 , and symmetric vectorization svec isdefined by (56). 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) (2074)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 (2075)Projection of P 2 X i ∑ j λ j q j q T j on the positive semidefinite cone (7.1.2) isfound from its eigenvalue decomposition (A.5.1);P 1 P 2 X i =n∑max{0 , λ j }q j qj T (2076)j=1