v2009.01.01 - Convex Optimization

264 CHAPTER 4. SEMIDEFINITE PROGRAMMING
problem (623) becomes equivalent to: (Theorem A.3.1.0.7)
minimize 1 Tˆx
X∈ S n , ˆx∈R n
subject to A(ˆx + 1) 1
[ = b 2
] X ˆx
G =
ˆx T 1
δ(X) = 1
(G ≽ 0)
rankG = 1
(625)
where solution is confined to rank-1 vertices of the elliptope in S n+1
(5.9.1.0.1) by the rank constraint, the positive semidefiniteness, and the
equality constraints δ(X)=1. The rank constraint makes this problem
nonconvex; by removing it 4.13 we get the semidefinite program
minimize 1 Tˆx
X∈ S n , ˆx∈R n
subject to A(ˆx + 1) 1
[ = b 2
] X ˆx
G =
ˆx T 1
δ(X) = 1
≽ 0
(626)
whose optimal solution x ⋆ (622) is identical to that of minimum cardinality
Boolean problem (614) if and only if rankG ⋆ =1. Hope 4.14 of acquiring a
rank-1 solution is not ill-founded because 2 n elliptope vertices have rank 1,
and we are minimizing an affine function on a subset of the elliptope
(Figure 106) containing rank-1 vertices; id est, by assumption that the
feasible set of minimum cardinality Boolean problem (614) is nonempty,
a desired solution resides on the elliptope relative boundary at a rank-1
vertex. 4.15
4.13 Relaxed problem (626) can also be derived via Lagrange duality; it is a dual of a
dual program [sic] to (625). [263] [53,5, exer.5.39] [329,IV] [125,11.3.4] The relaxed
problem must therefore be convex having a larger feasible set; its optimal objective value
represents a generally loose lower bound (1563) on the optimal objective of problem (625).
4.14 A more deterministic approach to constraining rank and cardinality is developed in
4.6.0.0.9.
4.15 Confinement to the elliptope can be regarded as a kind of normalization akin to
matrix A column normalization suggested in [101] and explored in Example 4.2.3.1.2.