Inexact generalized Newton methods for second order C-differentiable optimization

In this paper we define second order C-differentiable functions and second order C-differential operators, describe their some properties and propose an inexact generalized Newton method to solve unconstrained optimization problems in which the objective function is not twice differentiable, but second order C-differentiable. We prove that the algorithm is linearly convergent or superlinearly convergent including the case of quadratic convergence depending on various conditions on the objective function and different values for the control parameter in the algorithm.