Note that (6.8) is equivalent to the statement
that the vector
is in the column space of ,
or that there exists a free variable such that
As an aside, note that condition (6.8) also assures that (6.5)
will have zero derivative along the manifold, which is one of
the assumptions of the surrogate density.
The method to find the centroid is to find the
vector such that

(6.10) 
where
is equation (6.3) applied elementwise.
This is essentially the same as for the
positive case (5.18) , except that the nonlinear relationship
between
and
is different.
The algorithm of Section
5.3.2 to find
based on
driving (5.20) to zero can be used if
and
the diagonal elements of
in (5.21)
are given by

(6.11) 
Let
be the value of
at the solution to (6.10).
The modified algorithm to find
is:
 Set iteration counter .
 To initialize, let
.

. Initially,
will be the
vector of zeros.
 Compute
from
using (6.3) elementwise.
 Compute derivative and Hessian according to (5.21) and (5.23),
and (6.11), then update :
 Increment and go to step 3.
The above algorithm is implemented by software/lam_solve.m with dbound=1.
Baggenstoss
20170519