A First Order Sufficient Condition for Optimality in Nonsmooth Optimization

We derive a simple sufficient condition for a point a∈S⊂X be a local minimum of f: X R on S. This condition is of the first order in its nature and takes into account the derivative (or some generalization of it) in a neighborhood of a. Applications like a sufficient condition for (a, b)∈ S × T ⊂ X × Y be a saddle-point of a bivariate function f: X × Y ⊂ R are also proposed.