v2010.10.26 - Convex Optimization

354 CHAPTER 4. SEMIDEFINITE PROGRAMMINGThis is transformed to an equivalent convex problem by moving the rankconstraint to the objective: We iterate solution ofwithminimize 〈G , Y 〉X∈S N , x∈R Nsubject to Ax = b[X CxG =x T C T 1trX = 1minimizeY ∈ S N+1 〈G ⋆ , Y 〉subject to 0 ≼ Y ≼ ItrY = N]≽ 0(782)(783)until convergence. Direction matrix Y ∈ S N+1 , initially 0, regulates rank.(1700a) Singular value decomposition G ⋆ = UΣQ T ∈ S N+1+ (A.6) provides anew direction matrix Y = U(:, 2:N+1)U(:, 2:N+1) T that optimally solves(783) at each iteration. An optimal solution to (779) is thereby found in afew iterations, making convex problem (782) its equivalent.It remains possible for the iteration to stall; were a rank-1 G matrix notfound. In that case, the current search direction is momentarily reversedwith an added randomized element:Y = −U(:,2:N+1) ∗ (U(:,2:N+1) ′ + randn(N,1) ∗U(:,1) ′ ) (784)in Matlab notation. This heuristic is quite effective for problem (779) whichis exceptionally easy to solve by convex iteration.When b /∈R(A) then problem (779) must be restated as a projection:minimize ‖Ax − b‖x∈R Nsubject to ‖Cx‖ = 1(785)This is a projection of point b on an ellipsoid boundary because any affinetransformation of an ellipsoid remains an ellipsoid. Problem (782) in turnbecomesminimize 〈G , Y 〉 + ‖Ax − b‖X∈S N , x∈R N[ ]X Cxsubject to G =x T C T ≽ 0 (786)1trX = 1