  We prove that the finite-difference based derivative-free descent (FD-DFD)
methods have a capability to find the global minima for a class of multiple
minima problems. Our main result shows that, for a class of multiple minima
objectives that is extended from strongly convex functions with
Lipschitz-continuous gradients, the iterates of FD-DFD converge to the global
minimizer $x_*$ with the linear convergence $\|x_{k+1}-x_*\|_2^2\leqslant\rho^k
\|x_1-x_*\|_2^2$ for a fixed $0<\rho<1$ and any initial iteration
$x_1\in\mathbb{R}^d$ when the parameters are properly selected. Since the
per-iteration cost, i.e., the number of function evaluations, is fixed and
almost independent of the dimension $d$, the FD-DFD algorithm has a complexity
bound $\mathcal{O}(\log\frac{1}{\epsilon})$ for finding a point $x$ such that
the optimality gap $\|x-x_*\|_2^2$ is less than $\epsilon>0$. Numerical
experiments in various dimensions from $5$ to $500$ demonstrate the benefits of
the FD-DFD method.
