v2009.01.01 - Convex Optimization

278 CHAPTER 4. SEMIDEFINITE PROGRAMMING
N∑
λ(G ⋆ ) i
i=n+1
= minimize
W ∈ S N 〈G ⋆ , W 〉
subject to 0 ≼ W ≼ I
trW = N − n
(1581a)
whose feasible set is a Fantope (2.3.2.0.1), and where G ⋆ is an optimal
solution to problem (670) given some iterate W . The idea is to iterate
solution of (670) and (1581a) until convergence as defined in4.4.1.2. 4.22
Optimal matrix W ⋆ is defined as any direction matrix yielding optimal
solution G ⋆ of rank n or less to then convex equivalent (670) of feasibility
problem (669); id est, any direction matrix for which the last N − n
nonincreasingly ordered eigenvalues λ of G ⋆ are zero: (p.599)
N∑
λ(G ⋆ ) i = 〈G ⋆ , W ⋆ 〉 = ∆ 0 (671)
i=n+1
We emphasize that convex problem (670) is not a relaxation of the
rank-constrained feasibility problem (669); at convergence, convex iteration
(670) (1581a) makes it instead an equivalent problem. 4.23
4.4.1.1 direction interpretation
We make no assumption regarding uniqueness of direction matrix W . The
feasible set of direction matrices in (1581a) is the convex hull of outer product
of all rank-(N − n) orthonormal matrices; videlicet,
conv { UU T | U ∈ R N×N−n , U T U = I } = { A∈ S N | I ≽ A ≽ 0, 〈I , A 〉= N − n } (82)
Set {UU T | U ∈ R N×N−n , U T U = I} comprises the extreme points of this
Fantope (82). An optimal solution W to (1581a), that is an extreme point,
is known in closed form (p.599). By (205), that particular direction (−W)
can be regarded as pointing toward positive semidefinite rank-n matrices that
4.22 The proposed iteration is not an alternating projection. (confer Figure 140)
4.23 Terminology equivalent problem meaning, optimal solution to one problem can be
derived from optimal solution to another. Terminology same problem means: optimal
solution set for one problem is identical to the optimal solution set of another (without
transformation).