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We propose and analyze asymptotic proximal point (APP) methods to find theglobal minimizer for a class of nonconvex, nonsmooth, or even discontinuousmultiple minima functions. The method is based on an asymptotic representationof nonconvex proximal points so that it can find the global minimizer withoutbeing trapped in saddle points, local minima, or even discontinuities. Our mainresult shows that the method enjoys the global linear convergence for such aclass of functions. Furthermore, the method is derivative-free and itsper-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 findinga point such that the gap between this point and the global minimizer is lessthan $\epsilon>0$. Numerical experiments and comparisons in various dimensionsfrom $2$ to $500$ demonstrate the benefits of the method.

Asymptotic proximal point methods: finding the global minima with linear convergence for a class of multiple minima problems