v2009.01.01 - Convex Optimization

324 CHAPTER 4. SEMIDEFINITE PROGRAMMING
desired cardinality cardδ(X) , and Y to find an approximating rank-one
matrix X :
maximize 〈X , A − w 1 Y 〉 − w 2 〈δ(X) , δ(W)〉
X∈S N
subject to 〈X , I〉 = 1
X ≽ 0
(744)
where w 1 and w 2 are positive scalars respectively weighting tr(XY ) and
δ(X) T δ(W) just enough to insure that they vanish to within some numerical
precision, where direction matrix Y is an optimal solution to semidefinite
program
minimize
Y ∈ S N 〈X ⋆ , Y 〉
subject to 0 ≼ Y ≼ I
trY = N − 1
(745)
and where diagonal direction matrix W ∈ S N optimally solves linear program
minimize 〈δ(X ⋆ ) , δ(W)〉
W=δ 2 (W)
subject to 0 ≼ δ(W) ≼ 1
trW = N − c
(746)
Both direction matrix programs are derived from (1581a) whose analytical
solution is known but is not necessarily unique. We emphasize (confer p.278):
because this iteration (744) (745) (746) (initial Y,W = 0) is not a projection
method, success relies on existence of matrices in the feasible set of (744)
having desired rank and diagonal cardinality. In particular, the feasible set
of convex problem (744) is a Fantope (83) whose extreme points constitute
the set of all normalized rank-one matrices; among those are found rank-one
matrices of any desired diagonal cardinality.
Convex problem (744) is neither a relaxation of cardinality problem (740);
instead, problem (744) is a convex equivalent to (740) at convergence of
iteration (744) (745) (746). Because the feasible set of convex problem (744)
contains all normalized (B.1) symmetric rank-one matrices of every nonzero
diagonal cardinality, a constraint too low or high in cardinality c will
not prevent solution. An optimal rank-one solution X ⋆ , whose diagonal
cardinality is equal to cardinality of a principal eigenvector of matrix A ,
will produce the lowest residual Frobenius norm (to within machine noise
processes) in the original problem statement (739).