v2007.11.26 - Convex Optimization

4.5. CONSTRAINING CARDINALITY 2734.5 Constraining cardinality4.5.1 nonnegative variableOur goal is to reliably constrain rank in a semidefinite program. Thereis a direct analogy to linear programming that is simpler to present butequally hard to solve. In Optimization, that analogy is known as thecardinality problem. If we can solve the cardinality problem, then solutionto the rank-constraint problem follows; and vice versa.Consider a feasibility problem Ax = b , but with an upper bound k oncardinality ‖x‖ 0 of a nonnegative solution x : for vector b∈R(A)find x ∈ R nsubject to Ax = bx ≽ 0‖x‖ 0 ≤ k(646)where ‖x‖ 0 ≤ k means 4.27 vector x has at most k nonzero entries; sucha vector is presumed existent in the feasible set. Nonnegativity constraintx ≽ 0 is analogous to positive semidefiniteness; the notation means vector xbelongs to the nonnegative orthant R n + . Cardinality is quasiconcave on R n +just as rank is quasiconcave on S n + . [46,3.4.2]We propose that cardinality-constrained feasibility problem (646) isequivalently expressed with convex constraints:minimize x T yx∈R n , y∈R nsubject to Ax = bx ≽ 00 ≼ y ≼ 1y T 1 = n − k(647)whose bilinear objective function x T y is quasiconcave only when n = 1.This simple-looking problem (647) is very hard to solve, yet is not hardto understand. Because the sets feasible to x and y are not interdependent,4.27 Strictly speaking, cardinality ‖x‖ 0 cannot be a norm (3.1.3) because it is notpositively homogeneous.