v2010.10.26 - Convex Optimization

4.2. FRAMEWORK 293has norm ‖x ⋆ ‖ 2 =1 and minimal cardinality; the minimum number ofnonzero entries in vector x . The Matlab backslash command x=A\b ,for example, finds⎡x M=⎢⎣2128051280901280⎤⎥⎦(684)having norm ‖x M‖ 2 = 0.7044 .id est, an optimal solution toCoincidentally, x Mis a 1-norm solution;minimize ‖x‖ 1xsubject to Ax = b(518)The pseudoinverse solution (rounded)⎡ ⎤−0.0456−0.1881x P= A † b =0.0623⎢ 0.2668⎥⎣ 0.3770 ⎦−0.1102(685)has least norm ‖x P‖ 2 =0.5165 ; id est, the optimal solution to (E.0.1.0.1)minimize ‖x‖ 2xsubject to Ax = b(686)Certainly none of the traditional methods provide x ⋆ = e 4 (683) because, andin general, for Ax = b∥ arg inf ‖x‖2∥∥2 ≤ ∥ ∥ arg inf ‖x‖1∥∥2 ≤ ∥ ∥ arg inf ‖x‖0∥∥2 (687)