## Method 1 (requiring starting point)

To compute the Hessian, we take the derivative of (6.7) with respect to ,

 (6.9)

The following algorithm is proposed
1. Determine the pseudo-inverse solution (5.11).
2. Seek an initial starting point for , a vector that meets (5.12) and has elements on . Use an off-the-shelf linear programming solver as explained in Section 5.3.1.
3. Compute free variable so that using    .
4. Determine from by solving (6.3) for , for each .
5. Compute entropy (6.6) and first and second derivatives (6.7),(6.9).
6. Take a Newton-Raphson iteration :

where and are the Hessian and gradient of with respect to .
7. Re-compute the mean : . Check that all elements of are in . If not, take a smaller step.
8. Go to step 4.

The above algorithm is implemented by software/me_lin_01.m.

Baggenstoss 2017-05-19