v2010.10.26 - Convex Optimization

4.5. CONSTRAINING CARDINALITY 351permit number of generators to be reduced below n by discerning thesmallest face. 4.37There is wondrous bonus to presolving when a constraint matrix is sparse.After columns are removed by theory of convex cones (finding the smallestface), some remaining rows may become 0 T , identical to other rows, ornonnegative. When nonnegative rows appear in an equality constraint to 0,all nonnegative variables corresponding to nonnegative entries in those rowsmust vanish (A.7.1); meaning, more columns may be removed. Once rowsand columns have been removed from a constraint matrix, even more rowsand columns may be removed by repeating the presolver procedure.4.5.2 constraining cardinality of signed variableNow consider a feasibility problem equivalent to the classical problem fromlinear algebra Ax = b , but with an upper bound k on cardinality ‖x‖ 0 : forvector b∈R(A)find x ∈ R nsubject to Ax = b‖x‖ 0 ≤ k(775)where ‖x‖ 0 ≤ k means vector x has at most k nonzero entries; such a vectoris presumed existent in the feasible set. Convex iteration (4.5.1) utilizes anonnegative variable; so absolute value |x| is needed here. We propose thatnonconvex problem (775) can be equivalently written as a sequence of convexproblems that move the cardinality constraint to the objective:minimizex∈R n 〈|x| , y〉subject to Ax = b≡minimize 〈t , y + ε1〉x∈R n , t∈R nsubject to Ax = b−t ≼ x ≼ t(776)minimizey∈R n 〈t ⋆ , y + ε1〉subject to 0 ≼ y ≼ 1y T 1 = n − k(525)4.37 But a canonical set of conically independent generators of K comprise only the firsttwo and last two columns of A.