A secant method for nonsmooth optimization

The notion of a secant for locally Lipschitz continuous functions is introduced and a new algorithm to locally minimize nonsmooth, nonconvex functions based on secants is developed. We demonstrate that the secants can be used to design an algorithm to find descent directions of locally Lipschitz continuous functions. This algorithm is applied to design a minimization method, called a secant method. It is proved that the secant method generates a sequence converging to Clarke stationary points. Numerical results are presented demonstrating the applicability of the secant method in wide variety of nonsmooth, nonconvex optimization problems. We also compare the proposed algorithm with the bundle method using numerical results.