v2009.01.01 - Convex Optimization

5.4. EDM DEFINITION 359
5.4.2.2.1 Example. List-member constraints via Gram matrix.
Capitalizing on identity (820) relating Gram and EDM D matrices, a
constraint set such as
tr ( − 1V DV e )
2 ie T i = ‖xi ‖ 2
⎫⎪
tr ( ⎬
− 1V DV (e 2 ie T j + e j e T i ) 2) 1 = x
T
i x j
tr ( (826)
− 1V DV e ) ⎪
2 je T j = ‖xj ‖ 2 ⎭
relates list member x i to x j to within an isometry through inner-product
identity [332,1-7]
cos ψ ij =
xT i x j
‖x i ‖ ‖x j ‖
(827)
where ψ ij is angle between the two vectors as in (807). For M list members,
there total M(M+1)/2 such constraints. Angle constraints are incorporated
in Example 5.4.2.2.2 and Example 5.4.2.2.9.
Consider the academic problem of finding a Gram matrix subject to
constraints on each and every entry of the corresponding EDM:
find
D∈S N h
−V DV 1 2 ∈ SN
subject to 〈 D , (e i e T j + e j e T i ) 1 2〉
= ď ij , i,j=1... N , i < j
−V DV ≽ 0
(828)
where the ďij are given nonnegative constants. EDM D can, of course,
be replaced with the equivalent Gram-form (810). Requiring only the
self-adjointness property (1321) of the main-diagonal linear operator δ we
get, for A∈ S N
〈D , A〉 = 〈 δ(G)1 T + 1δ(G) T − 2G , A 〉 = 〈G , δ(A1) − A〉 2 (829)
Then the problem equivalent to (828) becomes, for G∈ S N c ⇔ G1=0
find
G∈S N c
subject to
G ∈ S N
〈
G , δ ( (e i e T j + e j e T i )1 ) 〉
− (e i e T j + e j e T i ) = ďij , i,j=1... N , i < j
G ≽ 0 (830)