I would use a genetic algorithm to deal with the integer parameter, at least.

I guess this is a duplicate of another question with already several answers. But in case the question is still open, I leave here my contribution:

Your problem seems to be called a “tuning problem”.

You can treat the integer parameter as a continuous real one and run the optimization under that hypothetical scenario. Hopefully, along the way, the value in that parameter does not stand between -1 and 1 (in several scenarios that might determine big changes in the values). Given the final value for the parameter in the optimum, you can round it back to the nearest integer.

Then you can evaluate the function for a range of integers below and above that value (say, for example, variations between -10 and +10) to see how far it goes from the optimum. This would be equivalent to perform a sensitivity analysis. If the optimum holds or varies smoothly and regularly, then probably you have the optimum as the minimum or maximum obtained among the several tries for that range of values.

Of course that one should pay attention to what happens with that integer parameter in the function, to make sure that the assumed (and partly checked) similarity of the integer with the real continuous equivalent is not misguiding. Probably some scenarios could be designed that would make the idea fail.

For a more rigorous approach, apparently grounded on this idea, see “Continuous optimization algorithms for tuning real and integer parameters of swarm intelligence algorithms” by Zhi Yuan, Marco A. Montes de Oca, Thomas Stutzle, and Mauro Birattari, in http://iridia.ulb.ac.be/IridiaTrSeries/rev/IridiaTr2011-017r001.pdf

In this paper - that supports the idea - the domain of the integer parameters is treated as a continuous domain by the continuous optimization algorithms, and the returned values are then rounded to the nearest integer. This approach is reasonable, when the domain of an integer parameter is large.