v2009.01.01 - Convex Optimization

4.5. CONSTRAINING CARDINALITY 295
minimize
x∈R n 〈x , y〉
subject to Ax = b
x ≽ 0
(143)
n∑
π(x ⋆ ) i =
i=k+1
minimize
y∈R n 〈x ⋆ , y〉
subject to 0 ≼ y ≼ 1
y T 1 = n − k
(467)
where π is the (nonincreasing) presorting function. This sequence is iterated
until x ⋆T y ⋆ vanishes; id est, until desired cardinality is achieved.
Problem (467) is analogous to the rank constraint problem; (Ky Fan,
confer p.278)
N∑
λ(G ⋆ ) i
i=k+1
= minimize
W ∈ S N 〈G ⋆ , W 〉
subject to 0 ≼ W ≼ I
trW = N − k
(1581a)
Linear program (467) sums the n−k smallest entries from vector x . In
context of problem (682), we want n−k entries of x to sum to zero; id est,
we want a globally optimal objective x ⋆T y ⋆ to vanish: more generally,
n∑
π(|x ⋆ |) i = 〈|x ⋆ |, y ⋆ 〉 = ∆ 0 (683)
i=k+1
Then n−k entries of x ⋆ are themselves zero whenever their absolute sum
is, and cardinality of x ⋆ ∈ R n is at most k .
4.5.1.1 direction vector interpretation
Vector y may be interpreted as a negative search direction; it points opposite
to direction of movement of hyperplane {x | 〈x , y〉= τ} in a minimization of
real linear function 〈x , y〉 over the feasible set in linear program (143). (p.69)
Direction vector y is not unique. The feasible set of direction vectors in (467)
is the convex hull of all cardinality-(n−k) one-vectors; videlicet,
conv{u∈ R n | cardu = n−k , u i ∈ {0, 1}} = {a∈ R n | 1 ≽ a ≽ 0, 〈1, a〉= n−k}
(684)