v2009.01.01 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 289
which is a convex relaxation of the desired equality constraint
[ ] I X
X T =
G
[ I
X T ]
[ I X ] (843)
The rank constraint insures this equality holds thus restricting solution to R n .
convex equivalent problem statement
Problem statement (677) is nonconvex because of the rank constraint. We
do not eliminate or ignore the rank constraint; rather, we find a convex way
to enforce it: for 0 < n < N
minimize 〈Z , W 〉
G∈S N , X∈R n×N
subject to d ij ≤ 〈G , (e i − e j )(e i − e j ) T 〉 ≤ d ij ∀(i,j)∈ I
〈G , e i e T i 〉 = ‖ˇx i ‖ 2 , i = N − m + 1... N
〈G , (e i e T j + e j e T i )/2〉 = ˇx T i ˇx j , i < j , ∀i,j∈{N − m + 1... N}
X(:, N − m + 1:N) = [ ˇx N−m+1 · · · ˇx N ]
[ ] I X
Z =
X T
≽ 0 (678)
G
Each linear equality constraint in G∈ S N represents a hyperplane in
isometrically isomorphic Euclidean vector space R N(N+1)/2 , while each linear
inequality pair represents a convex Euclidean body known as slab (an
intersection of two parallel but opposing halfspaces, Figure 11). In this
convex optimization problem (678), a semidefinite program, we substitute
a vector inner-product objective function for trace from nonconvex problem
(677);
〈Z , I 〉 = trZ ← 〈Z , W 〉 (679)
a generalization of the known trace heuristic [113] for minimizing convex
envelope of rank, where W ∈ S N+n
+ is constant with respect to (678).
Matrix W is normal to a hyperplane in S N+n minimized over a convex feasible
set specified by the constraints in (678). Matrix W is chosen so −W points
in the direction of a feasible rank-n Gram matrix. Thus the purpose of vector
inner-product objective (679) is to locate a feasible rank-n Gram matrix that
is presumed existent on the boundary of positive semidefinite cone S N + .