v2007.09.17 - Convex Optimization

274 CHAPTER 4. SEMIDEFINITE PROGRAMMINGthe sparsest solution to the classical linear equation Ax = b is x = e 4 ∈ R 6(confer (591)). And given data, from Example 4.2.3.0.3,[ ] [ ]1 0 √2 11A =1, b = (595)0 1 √2 1the most sparse solution is x = [ 0 0 √ 2 ] T ∈ R 3 (confer (596)).random data, in Matlab notation,GivenA=randn(m,n), index=round((n−1)∗rand(1)) +1, b=A(:,index)(647)where m and n are selected arbitrarily, the sparsest solution is x=e index ∈ R nfrom the standard basis. Although these sparsest solutions are recoverableby inspection, we seek to discern them instead by convex iteration; namely,by iterating problem sequence (646) (434). From the numerical data given,cardinality ‖x‖ 0 = 1 is expected. Iteration continues until x T y vanishes (towithin some numerical precision); id est, until desired cardinality is achieved.All three examples return a correct cardinality-1 solution to withinmachine precision in few iterations, but are occasionally subject to stall.Stalls are remedied by reinitializing y to a random state. Stalling is not an inevitable behavior. Convex iteration succeeds, for sometypes of problem, all the time:4.4.3.0.2 Example. Projection on ellipsoid boundary. [38] [99,5.1][180,2] This problem is exceptionally easy to solve by convex iteration:Consider classical linear equation Ax = b but with a constraint on norm ofsolution x , given matrices C , A , and vector b∈R(A)find x ∈ R Nsubject to Ax = b‖Cx‖ = 1(648)The set {x | ‖Cx‖=1} describes an ellipsoid boundary. This is a nonconvexproblem because solution is constrained to that boundary. Assign[ ] [ ] [ ]Cx [ xG =T C T 1] X Cx ∆ Cxx =1x T C T =T C T Cx1 x T C T ∈ S N+11(649)