Multiplier Rules Under Mixed Assumptions of Differentiability and Lipschitz Continuity

In this paper we study nonlinear programming problems with equality, inequality, and abstract constraints where some of the functions are Fréchet differentiable at the optimal solution, some of the functions are Lipschitz near the optimal solution, and the abstract constraint set may be nonconvex. We derive Fritz John type and Karush–Kuhn–Tucker (KKT) type first order necessary optimality conditions for the above problem where Fréchet derivatives are used for the differentiable functions and subdifferentials are used for the Lipschitz continuous functions. Constraint qualifications for the KKT type first order necessary optimality conditions to hold include the generalized Mangasarian–Fromovitz constraint qualification, the no nonzero abnormal multiplier constraint qualification, the metric regularity of the constraint region, and the calmness constraint qualification.