v2009.01.01 - Convex Optimization

528 CHAPTER 7. PROXIMITY PROBLEMS
We can transform this problem to an equivalent Schur-form semidefinite
program; (3.1.7.2)
minimize
t∈R , D
subject to
t
[
]
tI vec(D − H)
vec(D − H) T ≽ 0 (1295)
1
D ∈ EDM N
characterized by great sparsity and structure. The advantage of this SDP
is lack of conditions on input H ; e.g., negative entries would invalidate any
solution provided by (1291). (7.0.1.2)
7.3.1.3 Gram-form semidefinite program, Problem 3 convex case
Further, this problem statement may be equivalently written in terms of a
Gram matrix via linear bijective (5.6.1) EDM operator D(G) (810);
minimize
G∈S N c , t∈R
subject to
t
[
tI vec(D(G) − H)
vec(D(G) − H) T 1
]
≽ 0
(1296)
G ≽ 0
To include constraints on the list X ∈ R n×N , we would rewrite this:
G∈S N c
minimize
, t∈R , X∈ Rn×N
t
subject to
[
tI vec(D(G) − H)
vec(D(G) − H) T 1
[ ] I X
X T ≽ 0
G
]
≽ 0
(1297)
X ∈ C
where C is some abstract convex set. This technique is discussed in5.4.2.2.4.