I have a nonconvex optimization problem with a linear objective function, a set of linear constraints and a set of nonlinear, non-convex constraints. Is this problem NP-hard? If so, how can I prove this?
In general, you can show that a class of problems is NP-Hard by taking a known NP-hard problem and reducing it to a problem in your class (being careful that size of the problem does not increase too much.)
Since some well known NP-hard problems can easily be rewritten as nonlinear optimization problems with non-convex constraints, the class of non-convex nonlinear optimization problems is in general NP-Hard.
However, this says nothing about your particular non-convex optimization problem.