Correct BFGS update

[timothy-nunn]

Having looked again into the maths behind VMCON, @mkovari pointed out that the implementation of the hessian matrix revision may be incorrect.

From the Crane report

(NOTE: the denominator of the second term should be $\xi^T\gamma$)

The calculation of $\gamma\gamma^T$ and $\xi\xi^T$ are doing an outer product which is different from the current implementation.

I have corrected the update formula and added a test that uses reference values from the original Crane implementation. These tests did not pass on our old implementation (using matrix multiplication) but do pass following my changes in vmcon.py (to use np.outer).

[mkovari]

Have you checked whether B is symmetric?

[mkovari]

Can you say something about what tests you have carried out? I can't seem to learn anything from the "Checks" tab.

[timothy-nunn]

Have you checked whether B is symmetric?

Within the tests I have added, there is a check for symmetry (up to 14 dp)

Can you say something about what tests you have carried out? I can't seem to learn anything from the "Checks" tab.

Most of these checks assert that PyVMCON is able to solve the example problems in the Crane report and produces (near) identical numeric results. I have also checked and this changes causes no differences to any of the PROCESS regression tests except for one (ST) that converges on a slightly different optima.

[mkovari]

@jonmaddock I would be interested to see what effect this has on your convergence studies.