Thread: Solving likelihood function with multiple variables

I am new to this forum so apologies in advance if I am in the wrong place.

I am trying to recreate the model described in the attached paper but need some help with some of the more advanced statistical methods employed.

In summary, the objective is to take a batch of historical soccer results data and use this information to predict the results of future games. The number of goals scored by either team can generally be described by independent Poisson variables, save for specific cases (i.e. when both teams score 1 goal or less). The basic Poisson model can be modified to account for the departure from independence for low scoring games, and then the likelihood function is maximized to estimate the various input parameters (attack and defense parameter of either team, home advantage parameter etc) .

This is where I get stuck - I don't know how to perform the numerical maximization of equation 4.3 on page 271 (the 7th page of the attached document).

If you are interested in helping/providing guidance/being involved in the project then please let me know.