On optimality conditions and duality results in a class of nonconvex quasidifferentiable optimization problems

In the paper, the class of nonconvex nonsmooth optimization problems with the quasidifferentiable functions is considered. Further, a new notion of nonsmooth generalized convexity, namely, the concept of r-invexity with respect to a convex compact set is introduced. Several conditions for quasidifferentiable r-invexity with respect to a convex compact set are given. Furthermore, the sufficient optimality conditions and several Mond-Weir duality results are established for the considered nonconvex quasidifferentiable optimization problem under assumption that the functions constituting it are r-invex with respect to the same function eta and with respect to convex compact sets which are equal to Minkowski sum of their subdifferentials and superdifferentials. It is also illustrated that, for such nonsmooth extremum problems, the Lagrange multipliers may not be constant.