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The presence of Lipschitzian properties for solution mappings associated withnonlinear parametric optimization problems is desirable in the context ofstability analysis or bilevel optimization. An example of such a Lipschitzianproperty for set-valued mappings, whose graph is the solution set of a systemof nonlinear inequalities and equations, is R-regularity. Based on theso-called relaxed constant positive linear dependence constraint qualification,we provide a criterion ensuring the presence of the R-regularity property. Inthis regard, our analysis generalizes earlier results of that type whichexploited the stronger Mangasarian-Fromovitz or constant rank constraintqualification. Afterwards, we apply our findings in order to derive newsufficient conditions which guarantee the presence of R-regularity for solutionmappings in parametric optimization. Finally, our results are used to derive anexistence criterion for solutions in pessimistic bilevel optimization and asufficient condition for the presence of the so-called partial calmnessproperty in optimistic bilevel optimization.

R-regularity of set-valued mappings under the relaxed constant positive  linear dependence constraint qualification with applications to parametric  and bilevel optimization