v2009.01.01 - Convex Optimization

4.1. CONIC PROBLEM 253
Proposition 2.9.3.0.1, the upper bound on rank of X existent in the feasible
set P is satisfied by an extreme point. The upper bound on rank of an
optimal solution X ⋆ existent in F(P) is thereby also satisfied by an extreme
point of P precisely because {X ⋆ } constitutes F(P) ; 4.4 in particular,
{X ⋆ ∈ P | rankX ⋆ ≤ 1} = {Γ 1 , Γ 2 } ⊆ F(P) (593)
As all linear functions on a polyhedron are minimized on a face, [82] [217]
[238] [241] by analogy we so demonstrate coexistence of optimal solutions X ⋆
of (584P) having assorted rank.
4.1.1.4 Previous work
Barvinok showed [23,2.2] when given a positive definite matrix C and an
arbitrarily small neighborhood of C comprising positive definite matrices,
there exists a matrix ˜C from that neighborhood such that optimal solution
X ⋆ to (584P) (substituting ˜C) is an extreme point of A ∩ S n + and satisfies
upper bound (245). 4.5 Given arbitrary positive definite C , this means
nothing inherently guarantees that an optimal solution X ⋆ to problem (584P)
satisfies (245); certainly nothing given any symmetric matrix C , as the
problem is posed. This can be proved by example:
4.1.1.4.1 Example. (Ye) Maximal Complementarity.
Assume dimension n to be an even positive number. Then the particular
instance of problem (584P),
minimize
X∈ S n 〈[ I 0
0 2I
subject to X ≽ 0
〈I , X〉 = n
] 〉
, X
(594)
has optimal solution
X ⋆ =
[ 2I 0
0 0
]
∈ S n (595)
4.4 and every face contains a subset of the extreme points of P by the extreme
existence theorem (2.6.0.0.2). This means: because the affine subset A and hyperplane
{X | 〈C , X 〉 = f ⋆ 0 } must intersect a whole face of P , calculation of an upper bound on
rank of X ⋆ ignores counting the hyperplane when determining m in (245).
4.5 Further, the set of all such ˜C in that neighborhood is open and dense.