Invexity of supremum and infimum functions

Under suitable assumptions we establish the formulas for calculating generalised gradients and generalised directional derivatives in the Clarke sense of the supremum and the infimum of an infinite family of Lipschitz functions. From these results we derive the results ensuring such a supremum or infimum are an invex function when all functions of the invex. Applying these results to a class of mathematical programs, we obtain necessary and sufficient conditions for optimality.