  We propose and analyze asymptotic proximal point (APP) methods to find the
global minimizer for a class of nonconvex, nonsmooth, or even discontinuous
multiple minima functions. The method is based on an asymptotic representation
of nonconvex proximal points so that it can find the global minimizer without
being trapped in saddle points, local minima, or even discontinuities. Our main
result shows that the method enjoys the global linear convergence for such a
class of functions. Furthermore, the method is derivative-free and its
per-iteration cost, i.e., the number of function evaluations, is also bounded,
so it has a complexity bound $\mathcal{O}(\log\frac{1}{\epsilon})$ for finding
a point such that the gap between this point and the global minimizer is less
than $\epsilon>0$. Numerical experiments and comparisons in various dimensions
from $2$ to $500$ demonstrate the benefits of the method.
