# Solving nonlinear optimization problem with combinational constraints

I have to minimize a nonlinear objective function $f(x_0, x_1, x_2, x_3, x_4, x_5)$ with 6 variables.

The constraints governing these these variables are a mix of nonlinear inequality constraints, and some combinatorial constraints.

For the nonlinear inequality constraints and the objective function, I can evaluate the functional and Jacobian values as long as all the variables ($x_0$ to $x_5$) are specified. I can also obtain the Hessian values if I want, but I afraid that the values might be inaccurate.

Also, the objective function and the nonlinear inequality constraints are always continuous, but only piecewise differentiable over the region of interest.

By combinatorial constraints, I mean that ${x_3, x_4, x_5}$ can only take a list of discrete values together, such as {3,4,5}, {8,7,18} and so on.

Is there any existing scientific package ( or mathematical algorithm, I can program the solver myself) that does what I need?

Note: the inequality constraints are highly nonlinear and can only be evaluated via numerical means.