Better optimisation method for Dixon-Coles model

[Torvaney]

Use an optimisation routine that enforces the constraints that make the Dixon-Coles model identifiable.

Ideally this would allow arbitrary constraints to be added to additional predictor variables specified in dixoncoles_ext.

[Torvaney]

I think optim is actually kind of okay (aside from warnings resulting from invalid rho estimates).

Currently I'm thinking a method kind of like this might work well:

• Specify constraints as a list of formulae. Something like c(x) ~ equal_to(0).
  • Could specify the penalty/barrier method here. Like gte_zero('log_barrier') or something.
• Convert formulae to a set of penalty functions.
• Create a new objective function as sum of original and the penalty functions.

I guess there could be some difficulties/annoyances with the penalty weights. These would probably need to be customisable either through some control type function argument (or varargs with!!!) or in the constraint/formula specification (like 0.1 * c(x) ~ equal_to(0)).

Sequential solutions with increasing penalty weights should also be considered. In this case each constraint could return a function that takes a weight and returns a penalty function (or just takes weight as an argument - maybe this is more idiomatic in R?).

[Torvaney]

Another thought: adding constraints to team parameters could be annoying.

One potential solution would be to look for parameters named by off___* or def___*. This would at least work for the default method, even if it is a little inelegant.

[Torvaney]

The easiest way to make model fitting faster is probably to compute the gradient of the the objective function, rather than relying on gradient-free methods...

[Torvaney]

Potentially a dumb idea, but given it's just matrices, could try any use the rstudio/keras api??

[Torvaney]

Closing for now since the kind of optimisation improvements I want would be their own project...