v2007.09.17 - Convex Optimization

272 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.4.2 cardinalityOur 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 equivalent to the classical problem fromlinear algebra Ax = b , but with an upper bound k on cardinality ‖x‖ 0 ofa nonnegative solution x : for vector b∈R(A)find x ∈ R nsubject to Ax = bx ≽ 0‖x‖ 0 ≤ k(644)where ‖x‖ 0 ≤ k means vector x has at most k nonzero entries; such a vectoris presumed existent in the feasible set. Nonnegativity constraint x ≽ 0 isanalogous to positive semidefiniteness; the notation means vector x belongsto the nonnegative orthant R n + . Cardinality is quasiconcave on R n + just asrank is quasiconcave on S n + . [46,3.4.2]We propose that cardinality-constrained feasibility problem (644) isequivalently expressed with convex constraints:minimize x T yx∈R n , y∈R nsubject to Ax = bx ≽ 00 ≼ y ≼ 1y T 1 = n − k(645)whose bilinear objective function x T y is quasiconcave only when n = 1.This simple-looking problem (645) is very hard to solve, yet is not hardto understand. Because the sets feasible to x and y are not interdependent,we can separate the problem half in variable y :minimize x T yy∈R nsubject to 0 ≼ y ≼ 1y T 1 = n − k(434)