# Optimal numerical method for optimization of “Rosenbrock Banana”-like function

Which numerical methods would be optimal to find an extremum of a function with an almost flat "valley" (but a single minimum in the middle of the valley)? In this context optimal means the least number of function evaluations.

For simplicity's sake, assume that it's a two-dimensional problem. Also, of importance may be that I am more interested not in the coordinates of the extremum, but the value of the function in it.

For reference, the Rosenbrock function is $$f(x,y) = (1-x)^2 + 100(y-x^2)^2$$

I am using the term "'Rosenbrock'-like", because I have not yet defined the function itself. The problem I am solving is that of nonlinear equality constraint optimization, which I believe I have reduced to a global optimization of a function with a "valley" along the constraint.

• Is the objective function actually a sum of squares? Is objective function at least twice continously differentiable? – Brian Borchers Jun 2 '14 at 19:52
• @BrianBorchers It is not a sum of squares... but I do think it can be assumed to be sufficiently smooth. – Aahz Jun 2 '14 at 20:07
• Can you compute the gradient of the objective function reasonably efficiently? What about the Hessian? – Brian Borchers Jun 2 '14 at 20:31
• @BrianBorchers Well, as I have said, I am yet to define the precise function... But in general - I think we can compute the gradient (and the Hessian). – Aahz Jun 3 '14 at 5:04