https://www.mathworks.com/help/optim/ug/choosing-the-algorithm.html
Yes it can. I figure that the matrix size needed for you would be about 4 megabytes, so I think you should be able to use even midscale algorithms.
What you need to do is create an options structure that sets maximum function evaluations and maximum iterations much higher, and pass the options to fmincon. (The exact name of the options depends which of the options creation functions you use.)
