  This paper addresses the study of derivative-free smooth optimization
problems, where the gradient information on the objective function is
unavailable. Two novel general derivative-free methods are proposed and
developed for minimizing such functions with either global or local Lipschitz
continuous gradients. The newly developed methods use gradient approximations
based on finite differences, where finite difference intervals are
automatically adapted to the magnitude of the exact gradients without knowing
them exactly. The suggested algorithms achieve fundamental convergence results,
including stationarity of accumulation points in general settings as well as
global convergence with constructive convergence rates when the Kurdyka-\L
ojasiewicz property is imposed. The local convergence of the proposed
algorithms to nonisolated local minimizers, along with their local convergence
rates, is also analyzed under this property. Numerical experiences involving
various convex, nonconvex, noiseless, and noisy functions demonstrate that the
new methods exhibit essential advantages over other state-of-the-art methods in
derivative-free optimization.
