Nondifferentiable optimization by smooth approximations

Abstract

Necessary Lagrangian conditions are obtained for nonsmooth optimization problems, involving locally Lipschitz functions and arbitrary convex cones.The approach is by a sequence of smooth local approximations to the problem, which leads to Clarke generalized gradients in the limit.The method is applied to finite dimensional problems with convex cones, semiinfinite programming problems, continuous programming problems, and optimal control, giving Pontbyaoin type conditions.A duality theorem is also obtained, with the usual convexity hypotheses weakened to a generalized invex condition, using generalized gradients.