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Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Here, nonsmooth approximate gradient projection and complementary approximate Karush-Kuhn-Tucker conditions are presented. These sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constraint qualifications. It is proved that nonsmooth complementary approximate Karush-Kuhn-Tucker conditions are stronger than nonsmooth approximate gradient projection conditions. Sufficiency for differentiable generalized convex programming is established.

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‎In this paper, using the idea of convexificators, we study boundedness and nonemptiness of Lagrange multipliers satisfying the first order necessary conditions. We consider a class of nons- mooth fractional programming problems with equality, inequality constraints and an arbitrary set constraint. Within this context, define generalized Mangasarian-Fromovitz constraint qualification and show that the constraint qualification are necessary and suficient conditions for the Karush- Kuhn-Tucker(KKT) multipliers set to be nonempty and bounded.