v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 323
4.6.0.0.9 Example. Cardinality/rank problem.
d’Aspremont et alii [83] propose approximating a positive semidefinite matrix
A∈ S N + by a rank-one matrix having a constraint on cardinality c : for
0 < c < N
minimize ‖A − zz T ‖ F
z
(739)
subject to cardz ≤ c
which, they explain, is a hard problem equivalent to
maximize x T Ax
x
subject to ‖x‖ = 1
cardx ≤ c
(740)
where z ∆ = √ λx and where optimal solution x ⋆ is a principal eigenvector
(1575) (A.5) of A and λ = x ⋆T Ax ⋆ is the principal eigenvalue [134, p.331]
when c is true cardinality of that eigenvector. This is principal component
analysis with a cardinality constraint which controls solution sparsity. Define
the matrix variable
X ∆ = xx T ∈ S N (741)
whose desired rank is 1, and whose desired diagonal cardinality
cardδ(X) ≡ cardx (742)
is equivalent to cardinality c of vector x . Then we can transform cardinality
problem (740) to an equivalent problem in new variable X : 4.38
maximize
X∈S N 〈X , A〉
subject to 〈X , I〉 = 1
(X ≽ 0)
rankX = 1
cardδ(X) ≤ c
(743)
We transform problem (743) to an equivalent convex problem by
introducing two direction matrices into regularization terms: W to achieve
4.38 A semidefiniteness constraint X ≽ 0 is not required, theoretically, because positive
semidefiniteness of a rank-1 matrix is enforced by symmetry. (Theorem A.3.1.0.7)