The presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.

与非线性参数优化问题相关的解映射的 Lipschitzian 属性的存在在例如稳定性分析或双层优化的上下文中是可取的。集合值映射的这种 Lipschitzian 性质的一个例子是 R 正则性，其图是非线性不等式和方程组的解集。基于所谓的宽松常数正线性相关约束条件，我们提供了一个确保 R 正则性存在的标准。在这方面，我们的分析概括了利用更强的 Mangasarian-Fromovitz 或恒定秩约束条件的早期结果。然后，我们应用我们的发现来推导出新的充分条件，以保证参数优化中解映射的 R 正则性的存在。最后，我们的结果用于推导出悲观双层优化中解的存在标准和乐观双层优化中所谓的部分平静属性存在的充分条件。