Are there any optimization algorithms aimed at finding a coinciding root and a (local) minimum of a multi-variable function
f. Say it is known analytically that these should coincide for the function.
I am interested in any such algorithm but more specifically for the case where
f is continuous, at least twice differentiable and very expensive to compute (so internal costs for the algorithm can be neglected, only function calls matter). What would you use if the gradient could be computed for free? What if the Hessian can also be computed for free? What if the root and the minimum only coincide approximately, i.e. the minimum $f(x_c)\in $ $[-\epsilon,\epsilon]$ for some small epsilon.