Abstract

A class of nonconvex optimization problems in which all the functions involved are directionally differentiable is considered. Necessary optimality conditions of Kuhn-Tucker type based on the directional derivatives are proved. Here the Lagrange multipliers generally depend on the directions. It is shown that for various concrete classes of problems (including classes convex problems, locally Lipschitz problems, composite nonsmooth problems), generalized Lagrange multipliers collapse to the standard ones (i.e., Lagrange multipliers are constants as usual). Optimality conditions for quasidifferentiable problems are derived from the main results. Optimality conditions for a class of problems in which all the functions possess upper DSL-approximates are also derived from the framework.