Geometric Programming with Exponentially Long Constraints

[Mia P]

I have a geometric programming problem with 100 variables and around 100 constraints. The objective function and most of the constraints are linear, but one of the constraints is a polynomial of degree n with an exponential number of terms so it cannot be written. However we do have functions that can output the value, derivative and hessian of the long constraint without writing it out. We would like the degree n to be as high as possible but would be happy with n up to 50. 

In our problem, it is very important that we are guaranteed to get the global optimum (or something that converges to it when the solver's runtime increases). 

1/ Can we solve this problem using CVXOPT? If not, are there other (preferably Matlab/Python) packages or softwares that we can use? 

2/ Calculating the value of the long constraint is fast, but calculating its derivative is moderately slow and calculating its Hessian could be very slow. Is there any package/solver that is guaranteed to get the global optimum without requiring the Hessian of the long constraint (or preferably, also without the derivative)?