Abstract
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.
Highlights
Lipschitzian properties of implicitly given set-valued mappings are of essential importance in order to study the stability of optimization problems, see e.g. [25, 32, 37] and the references therein
Our main results Theorems 3.2 and 3.3 depict that this is possible. With these new sufficient conditions for the presence of R-regularity for the mapping at hand, we are in position to state new criteria ensuring local Lipschitz continuity of the marginal function and R-regularity of the solution mapping associated with nonlinear parametric optimization problems whose feasible region is modeled with the aid of
Supposing that models the feasible region of a given parametric optimization problem, certain constraint qualifications need to be imposed on the images of in order to ensure that the associated Karush–Kuhn–Tucker conditions provide a necessary optimality condition
Summary
Lipschitzian properties of implicitly given set-valued mappings are of essential importance in order to study the stability of optimization problems, see e.g. [25, 32, 37] and the references therein. Our main results Theorems 3.2 and 3.3 depict that this is possible With these new sufficient conditions for the presence of R-regularity for the mapping at hand, we are in position to state new criteria ensuring local Lipschitz continuity of the marginal function and R-regularity of the solution mapping associated with nonlinear parametric optimization problems whose feasible region is modeled with the aid of. Afterwards, we use these findings in order to study the existence of so-called pessimistic solutions as well as the presence of the celebrated partial calmness property in bilevel optimization.
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