Suppose I have a energy functional E depending on X, where X is a N-dimensional real value vector and N could be very large (~=2000). I assume that there exists (at least) a (local) minimizer for E. There are several problems I am dealing with now:
Firstly, the functional E is extremely 'complicated', it is nearly impossible to compute its exact gradient and its Hessian as well. So the 'input' data are always approximated. (approximated Gradient by finite difference formula, approximate Hessian, ...)
Secondly, N is very large: N ~= 2000. So I guess this can be called a Large Scaled Problem.
Thirdly, in my code, I need to repeat this optimization problem many times (approx. 100 iterations), so a fast and efficient optimization solver is very necessary.
Is there any good algorithm that can satisfies all the requirement above?