## Method 2 (not requiring starting point)

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 element-wise. This is essentially the same as for the positive- case (5.18) , except that the non-linear 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:
1. Set iteration counter .
2. To initialize, let .
3. . Initially, will be the vector of zeros.
4. Compute from using (6.3) element-wise.
5. Compute derivative and Hessian according to (5.21) and (5.23), and (6.11), then update :

6. Increment and go to step 3.
The above algorithm is implemented by software/lam_solve.m with dbound=1.

Baggenstoss 2017-05-19