v2009.01.01 - Convex Optimization

294 CHAPTER 4. SEMIDEFINITE PROGRAMMING
We numerically tested the foregoing technique for constraining rank on
a wide range of problems including localization of randomized positions,
stress (7.2.2.7.1), ball packing (5.4.2.2.3), and cardinality problems. We
have had some success introducing the direction matrix inner-product (679)
as a regularization term (a multiobjective optimization) whose purpose is to
constrain rank, affine dimension, or cardinality:
4.5 Constraining cardinality
The convex iteration technique for constraining rank, discovered in 2005, was
soon applied to cardinality problems:
4.5.1 nonnegative variable
Our goal is to reliably constrain rank in a semidefinite program. There
is a direct analogy to linear programming that is simpler to present but,
perhaps, more difficult to solve. In Optimization, that analogy is known as
the cardinality problem.
Consider a feasibility problem Ax = b , but with an upper bound k on
cardinality ‖x‖ 0 of a nonnegative solution x : for A∈ R m×n and vector
b∈R(A)
find x ∈ R n
subject to Ax = b
x ≽ 0
‖x‖ 0 ≤ k
(682)
where ‖x‖ 0 ≤ k means 4.29 vector x has at most k nonzero entries; such
a vector is presumed existent in the feasible set. Nonnegativity constraint
x ≽ 0 is analogous to positive semidefiniteness; the notation means vector x
belongs to the nonnegative orthant R n + . Cardinality is quasiconcave on R n +
just as rank is quasiconcave on S n + . [53,3.4.2]
We propose that cardinality-constrained feasibility problem (682) is
equivalently expressed as a sequence of convex problems: for 0≤k ≤n−1
4.29 Although it is a metric (5.2), cardinality ‖x‖ 0 cannot be a norm (3.1.3) because it
is not positively homogeneous.