v2010.10.26 - Convex Optimization

348 CHAPTER 4. SEMIDEFINITE PROGRAMMINGcolumns of measurement matrix A , corresponding to any particularsolution (to (523)) of cardinality greater than 1, may be deprecated andthe problem solved again with those columns missing. Such columnsare recursively removed from A until a cardinality-1 solution is found.Either a solution to problem (523) is cardinality-1 or column indices of A ,corresponding to a higher cardinality solution, do not intersect that indexcorresponding to the cardinality-1 solution. When problem (523) is firstsolved, in the example of Figure 103, solution is cardinality-2 at the kissingpoint • on the indicated edge of simplex S . Imagining that the correspondingcardinality-2 face F has been removed, then the simplex collapses to a linesegment along one of the Cartesian axes. When that line segment kissesline A , then the cardinality-1 vertex-solution illustrated has been found. Asimilar argument holds for any orientation of line A and point of entry tosimplex S .Because signed compressed sensing problem (518) can be equivalentlyexpressed in a nonnegative variable, as we learned in Example 3.2.0.0.1(p.230), and because a cardinality-1 constraint in (518) transforms toa cardinality-1 constraint in its nonnegative equivalent (522), then thiscardinality-1 recursive reconstruction algorithm continues to hold for a signedvariable as in (518). This cardinality-1 reconstruction algorithm also holdsmore generally when affine subset A has any higher dimension n−m .Although it is more efficient to search over the columns of matrix A fora cardinality-1 solution known a priori to exist, a combinatorial approach,tables are turned when cardinality exceeds 1 :4.5.1.8 cardinality-k geometric presolverThis idea of deprecating columns has foundation in convex cone theory.(2.13.4.3) Removing columns (and rows) 4.35 from A∈ R m×n in a linearprogram like (523) (3.2) is known in the industry as presolving; 4.36 theelimination of redundant constraints and identically zero variables prior tonumerical solution. We offer a different and geometric presolver:4.35 Rows of matrix A are removed based upon linear dependence. Assuming b∈R(A) ,corresponding entries of vector b may also be removed without loss of generality.4.36 The conic technique proposed here can exceed performance of the best industrialpresolvers in terms of number of columns removed, but not in execution time. Thisgeometric presolver becomes attractive when a linear or integer program is not solvableby other means; perhaps because of sheer dimension.