I\'m looking to run a gradient descent optimization to minimize the cost of an instantiation of variables. My program is very computationally expensive, so I\'m looking for
One of the best respected libraries for this kind of optimization work is the NAG libraries. These are used all over the world in universities and industry. They're available for C / FORTRAN. They're very non-free, and contain a lot more than just minimisation functions - A lot of general numerical mathematics is covered.
Anyway I suspect this library is overkill for what you need. But here are the parts pertaining to minimisation: Local Minimisation and Global Minimization.