To add to what Alan has said...
I like to introduce the concept of the "basin of attraction". This is a region in your parameter space, that is specific to a given objective function, AND to the optimizer used. Effectively, for a given solution, the basin of attraction for that solution, using your chosen optimizer, is the set of points (X) that converge to the solution (Y). The various solutions (Y) need not be the globally optimal solution. They may even be points of divergence, where no local solution is found, but the optimizer decides to go there anyway.
A basin of attraction is not necessarily a convex set. It may even be dis-contiguous.
As well, the concept of basin of attraction applies to essentially all numerical optimization problems - root finding, min/max problems, etc.
So, the problem is, you need to start your choice of optimizer inside the basin of attraction for the solution you hope to find. Yeah, not always easy to do, since some basins may be terribly small.
Worse, MLE problems are often difficult. They often have domains where you get logs of negative numbers, causing the optimizer to fail rather unpleasantly. Or, you have domains where the objective is just so tiny as to cause underflows (why you want to work on the log-likelihood function).
So choosing a good start point tends to be a nasty thing. Of course, since you provide the data, it makes sense that you have an intelligent start point, based on your knowledge of the process. Hey! It is YOUR problem after all. Nobody knows it better than you.
Some optimizers are less sensitive to a poor choice of starting values. Partitioned Least Squares methods are good for some problems, because they allow you to reduce the dimensionality of the parameter space that must be searched. (They are not applicable to your problem of course.)
A choice of solver that tries to be robust to starting value problems is the family of stochastic solvers. This includes genetic algorithms, bee or ant colony optimizers, simulated annealing. A virtue of these tools is they try not to get stuck in a local minimum.
Another choice is the set of algorithms called global solvers. At the very least, start with a multi-start method. Thus choose a random sample of start points, and see where the solver ends up. This improves the odds that at least ONE of those start points lies inside the basin of attraction of the globally optimal solution.
