v2006.03.09 - Convex Optimization

370 CHAPTER 6. SEMIDEFINITE PROGRAMMING(874)minimize ‖x‖ 0xsubject to Ax = bx i ∈ {0, 1} ,i=1... n=minimize ‖x‖ 1xsubject to Ax = bx i ∈ {0, 1} ,(875)i=1... nare the same. The Boolean constraint makes the 1-norm problem nonconvex.Given data 6.11⎡−1 1 8 1 1 01 1 1A = ⎣ −3 2 82 31 1−9 4 84 9− 1 2 31− 1 4 9⎤ ⎡⎦ , b = ⎣11214⎤⎦ (876)the obvious and desired solution to the problem posed,x ⋆ = e 4 ∈ R 6 (877)has norm ‖x ⋆ ‖ 2 = 1 and minimum cardinality; the minimum number ofnonzero entries in vector x . Though solution (877) is obvious, the simplestnumerical method of solution is, more generally, combinatorial. The Matlabbackslash command x=A\b , for example, finds⎡x M=⎢⎣2128051280901280⎤⎥⎦(878)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(879)6.11 This particular matrix A is full-rank having three-dimensional nullspace (but thecolumns are not conically independent).