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The paper deals with perturbed nonlinear programming problems under the relaxed constant rank regularity condition. We study the relation of the relaxed constant rank regularity condition with the error bound property, the directional differentiability of the optimal value function, and necessary and sufficient second order optimality conditions.

We study set-valued mappings defined by solution sets of parametric systems of equalities and inequalities. We prove Lipschitz-like continuity of these mappings under relaxed constant rank constraint qualification.2010 Mathematics Subject Classification. 41A50, 46C05, 49K27, 52A07, 90C31.

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For the classical nonlinear program, two new relaxations of the Mangasarian-Fromovitz constraint qualification are discussed and their relationship with some standard constraint qualifications is examined. In particular, we establish the equivalence of one of these constraint qualifications with the recently suggested by Andreani et al. Constant rank of the subspace component constraint qualification. As an application, we make use of this new constraint qualification in the local analysis of the solution map to a parameterized equilibrium problem, modeled by a generalized equation.