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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? Please suggest me.

Thanks for all your helps.

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    $N=2000$ is not large scale by the standards of modern software for nonlinear optimization. What do you know about the convexity and smoothness of your function? – 2017-02-23
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    @BrianBorchers There is no guarantee that E is convex. However, E is smooth, but its form is extremely complicated and I cannot compute its exact formulation for Gradient and Hessian matrix. – 2017-02-23
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    I just pick randomly the number 2000, I thought it is big enough. But the more we increase N, the better result for my problem is. – 2017-02-23
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    Do you need a global minimum or will you be satisfied with any local minimum? – 2017-02-23
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    well, of course a global minimum would be best, but I will be satisfied with a local minimum as well, I need to have first a minimizer (local or global) to see what is going on in my scheme. – 2017-02-23
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    How expensive are function evaluations? How imprecise are the function evaluations? Can you get reasonably precise gradients? – 2017-02-23
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    gradient is approximated by a Voronoi tessellation, so I think it is reasonably precise, and better than just finite difference approximation. – 2017-02-23

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