Multiplier rules under mixed assumptions of differentiability and Lipschitz continuity

In this paper we study nonlinear programming problems with equality, inequa lity, and abstract constraints where some of the functions are Frechet diff erentiable at the optimal solution, some of the functions are Lipschitz nea r the optimal solution, and the abstract constraint set may be nonconvex. W e derive Fritz John type and Karush-Kuhn-Tucker (KKT) type first order nece ssary optimality conditions for the above problem where Frechet derivatives are used for the differentiable functions and subdifferentials are used fo r the Lipschitz continuous functions. Constraint qualifications for the KKT type first order necessary optimality conditions to hold include the gener alized Mangasarian-Fromovitz constraint qualification, the no nonzero abnor mal multiplier constraint qualification, the metric regularity of the const raint region, and the calmness constraint qualification.