The optimization problem that I am dealing with is very similar to this example. In brief, I have some decision variables that can take real values with lower/upper bounds, and some other variables that are constrained to be integers (which are then mapped to another value). The objective function and constraints are non-convex and non-linear. So far, the genetic algorithm is the one method that I have found, but I would like to know about other techniques (that are possibly better/faster) to approach this problem.
If you have the mathematical formulas for your problem (i.e., it's not black box), you can use a local MINLP solver, such as BONMIN, KNITRO, or MINOTAUR, or a deterministic global optimisation solver such as ANTIGONE, BARON, Couenne, or Octeract Engine.
If your problem is black box but you have access to derivatives, and function evaluations, I believe that KNITRO also supports callbacks and you can do the same in MINOTAUR but you'll probably need to edit the source code a bit.
The local optimisation solvers will be fast but might not find good/any solutions, whereas the deterministic global optimisation solvers are typically slower but much more likely to find a feasible point, and guarantee global optimality in a finite number of steps.
Try to take a look at these non-convex programming techniques:
DC Programming and DCA