Second Order Optimality Conditions in Nonsmooth Unconstrained Optimization

AMS subject classification: 49J52, 90C30.

Second order sufficient and necessary conditions are given for a nonsmooth function f defined in a Banach space to attain a minimum at a point in the interior of its domain. At the beginning sufficient conditions in terms of Riemann type derivatives are introduced. The considered examples suggest improvements to gain more efficiency. Consequently second order conditions based on generalized Riemann type finite difference are proved and their efficiency is shown. On this ground a generalized second order derivative is defined.