The fsolve function has three main algorithms (trust region, LM, IP) which all can be tried until a verifiable solution is obtained. I personally never forget that there is a high chance that no solution(s) exist (as also mentioned in the MATLAB documentation).

Alternatively, if you have access to, I suggest trying Solve[] or Reduce[] or NSolve[] in Mathematica.

You can probably code something on your own in MATLAB, based on smoothing methods for non-differentiable equations. In those methods the min or max operators are replaced by smooth operators that mimic the min/max ones through a positive parameter that you iteratively decrease towards zero, restarting each run with a successively smaller and smaller value of that parameter. Depending on the "gravity" of the non-smoothness, this process may take you fairly close to the actual solution before instabilities in the solver stops your further progress. 

@Giulio

My suggestin is that start from 2 variables , then go step by step with your problem.

However if any difficulty contact Matlab office .

Good luck

Prof Rath

Dear Giulio Vialetto 

I suggest you to use fsolve command. I used it and it has a good performance.

The other software exists which is suitable: EES (Engineering equation solver)