v2010.10.26 - Convex Optimization

230 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSwherecB 1 = {[I −I ]a | a T 1=c, a≽0} (521)Then (520) is equivalent tominimize cc∈R , x∈R n , a∈R 2nsubject to x = [I −I ]aa T 1 = ca ≽ 0Ax = b≡minimizea∈R 2n ‖a‖ 1subject to [A −A ]a = ba ≽ 0(522)where x ⋆ = [I −I ]a ⋆ . (confer (517)) Significance of this result:(confer p.384) Any vector 1-norm minimization problem may have itsvariable replaced with a nonnegative variable of the same optimalcardinality but twice the length.All other things being equal, nonnegative variables are easier to solve forsparse solutions. (Figure 71, Figure 72, Figure 100) The compressed sensingproblem (518) becomes easier to interpret; e.g., for A∈ R m×nminimize ‖x‖ 1xsubject to Ax = bx ≽ 0≡minimize 1 T xxsubject to Ax = bx ≽ 0(523)movement of a hyperplane (Figure 26, Figure 30) over a bounded polyhedronalways has a vertex solution [94, p.22]. Or vector b might lie on the relativeboundary of a pointed polyhedral cone K = {Ax | x ≽ 0}. In the lattercase, we find practical application of the smallest face F containing b from2.13.4.3 to remove all columns of matrix A not belonging to F ; becausethose columns correspond to 0-entries in vector x .3.2.0.0.2 Exercise. Combinatorial optimization.A device commonly employed to relax combinatorial problems is to arrangedesirable solutions at vertices of bounded polyhedra; e.g., the permutationmatrices of dimension n , which are factorial in number, are the extremepoints of a polyhedron in the nonnegative orthant described by anintersection of 2n hyperplanes (2.3.2.0.4). Minimizing a linear objectivefunction over a bounded polyhedron is a convex problem (a linear program)that always has an optimal solution residing at a vertex.