To compute the Hessian, we take the derivative of (6.7) with respect to ,
The following algorithm is proposed
The above algorithm is implemented by software/me_lin_01.m.
- Determine the pseudo-inverse solution
- 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.
- Compute free variable so that
by solving (6.3) for , for each .
- Compute entropy (6.6) and first and second derivatives (6.7),(6.9).
- Take a Newton-Raphson iteration :
are the Hessian and gradient
of with respect to .
- Re-compute the mean :
. Check that
all elements of
are in .
If not, take a smaller step.
- Go to step 4.