C1,1 functions and optimality conditions

In this work we provide a characterization of C1,1 functions on Rn (that is,diferentiable with locally Lipschitz partial derivatives) by means of second directional divided differences. In particular, we prove that the class of C1,1 functions is equivalent to the class of functions with bounded second directional divided diferences. From this result we deduce a Taylor's formula forthis class of functions and some optimality conditions. The characterizations and the optimality conditions proved by Riemann derivatives can be useful to write minimization algorithms; in fact, only the values of the function are required to compute second order conditions.