v2007.09.13 - Convex Optimization

276 CHAPTER 4. SEMIDEFINITE PROGRAMMINGWhen b /∈R(A) then problem (648) must be restated as a projection:minimize ‖Ax − b‖x∈R Nsubject to ‖Cx‖ = 1(654)This is a projection of point b on an ellipsoid boundary because any affinetransformation of an ellipsoid remains an ellipsoid. Problem (651) in turnbecomesminimize 〈G , Y 〉 + ‖Ax − b‖X∈S N , x∈R N[ ]X Cxsubject to G =x T C T ≽ 0 (655)1trX = 1We iterate this with calculation of direction matrix Y as before until a rank-1G matrix is found.4.4.3.0.3 Example. Tractable polynomial constraint.The ability to handle rank constraints makes polynomial constraints(generally nonconvex) transformable to convex constraints. All optimizationproblems having polynomial objective and polynomial constraints can bereformulated as a semidefinite program with a rank-1 constraint. [209]Suppose we require3 + 2x − xy ≤ 0 (656)Assign⎡ ⎤x [ x y 1]G = ⎣ y ⎦1=[ X zz T 1]∆=⎡⎣x 2 xy xxy y 2 yx y 1⎤⎦∈ S 3 (657)The polynomial constraint (656) is equivalent to the constraint set (B.1.0.2)tr(GA) ≤ 0[ X zG =z T 1(G ≽ 0)rankG = 1](658)