v2007.09.13 - Convex Optimization

242 CHAPTER 4. SEMIDEFINITE PROGRAMMINGfor example, findshaving norm ‖x M‖ 2 = 0.7044 .id est, an optimal solution to⎡x M=⎢⎣2128051280901280⎤⎥⎦(580)Coincidentally, x Mis a 1-norm solution;minimize ‖x‖ 1xsubject to Ax = bThe pseudoinverse solution (rounded)⎡ ⎤−0.0456−0.1881x P= A † b =0.0623⎢ 0.2668⎥⎣ 0.3770 ⎦−0.1102(581)(582)has least norm ‖x P‖ 2 =0.5165 ; id est, the optimal solution tominimize ‖x‖ 2xsubject to Ax = b(583)Certainly, none of the traditional methods provide x ⋆ = e 4 (579).We can reformulate this minimum cardinality Boolean problem (576) asa semidefinite program: First transform the variableso ˆx i ∈ {−1, 1} ; equivalently,x ∆ = (ˆx + 1) 1 2(584)minimize ‖(ˆx + 1) 1‖ ˆx2 0subject to A(ˆx + 1) 1 = b 2δ(ˆxˆx T ) = 1(585)