@misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {On Second-Order Optimality Conditionsfor Optimal Control Problems Governedby the Obstacle Problem}, series = {Optimization}, journal = {Optimization}, issn = {1029-4945}, doi = {10.1080/02331934.2020.1778686}, pages = {41}, abstract = {This paper is concerned with second-order optimality conditions for Tikhonov regularized optimal control problems governed by the obstacle problem. Using a simple observation that allows to characterize the structure of optimal controls on the active set, we derive various conditions that guarantee the local/global optimality of first-order stationary points and/or the local/global quadratic growth of the reduced objective function. Our analysis extends and refines existing results from the literature and also covers those situations where the problem at hand involves additional box-constraints on the control. As a byproduct, our approach shows in particular that Tikhonov regularized optimal control problems for the obstacle problem can be reformulated as state-constrained optimal control problems for the Poisson equation and that problems involving a subharmonic obstacle and a convex objective function are uniquely solvable. The paper concludes with three counterexamples which illustrate that rather peculiar effects can occur in the analysis of second-order optimality conditions for optimal control problems governed by the obstacle problem and that necessary second-order conditions for such problems may be hard to derive.}, language = {en} } @misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacle-Type Quasi-Variational Inequalities}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 26}, abstract = {This paper is concerned with the sensitivity analysis of a class of parameterized fixed-point problems that arise in the context of obstacle-type quasi-variational inequalities. We prove that, if the operators in the considered fixed-point equation satisfy a positive superhomogeneity condition, then the maximal and minimal element of the solution set of the problem depend locally Lipschitz continuously on the involved parameters. We further show that, if certain concavity conditions hold, then the maximal solution mapping is Hadamard directionally differentiable and its directional derivatives are precisely the minimal solutions of suitably defined linearized fixed-point equations. In contrast to prior results, our analysis requires neither a Dirichlet space structure, nor restrictive assumptions on the mapping behavior and regularity of the involved operators, nor sign conditions on the directions that are considered in the directional derivatives. Our approach further covers the elliptic and parabolic setting simultaneously and also yields Hadamard directional differentiability results in situations in which the solution set of the fixed-point equation is a continuum and a characterization of directional derivatives via linearized auxiliary problems is provably impossible. To illustrate that our results can be used to study interesting problems arising in practice, we apply them to establish the Hadamard directional differentiability of the solution operator of a nonlinear elliptic quasi-variational inequality, which emerges in impulse control and in which the obstacle mapping is obtained by taking essential infima over certain parts of the underlying domain, and of the solution mapping of a parabolic quasi-variational inequality, which involves boundary controls and in which the state-to-obstacle relationship is described by a partial differential equation.}, language = {en} } @misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 24}, abstract = {We prove that solution operators of elliptic obstacle-type variational inequalities (or, more generally, locally Lipschitz continuous functions possessing certain pointwise-a.e. convexity properties) are Newton differentiable when considered as maps between suitable Lebesgue spaces and equipped with the strong-weak Bouligand differential as a generalized set-valued derivative. It is shown that this Newton differentiability allows to solve optimal control problems with H1-cost terms and one-sided pointwise control constraints by means of a semismooth Newton method. The superlinear convergence of the resulting algorithm is proved in the infinite-dimensional setting and its mesh independence is demonstrated in numerical experiments. We expect that the findings of this paper are also helpful for the design of numerical solution procedures for quasi-variational inequalities and the optimal control of obstacle-type variational problems.}, language = {en} } @misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacle-Type Quasi-Variational Inequalities}, series = {SIAM Journal on Control and Optimization}, volume = {60}, journal = {SIAM Journal on Control and Optimization}, number = {6}, issn = {1095-7138}, doi = {10.1137/21m1419635}, pages = {3430 -- 3456}, language = {en} } @misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {Energy Space Newton Differentiability for Solution Maps of Unilateral and Bilateral Obstacle Problems}, series = {arXiv}, journal = {arXiv}, doi = {10.48550/arXiv.2308.15289}, pages = {1 -- 26}, language = {en} } @misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control}, series = {SIAM Journal on Control and Optimization}, volume = {61}, journal = {SIAM Journal on Control and Optimization}, number = {3}, issn = {1095-7138}, doi = {10.1137/21m1467365}, pages = {1162 -- 1186}, language = {en} } @misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {Energy Space Newton Differentiability for Solution Maps of Unilateral and Bilateral Obstacle Problems}, series = {SIAM Journal on Control and Optimization}, volume = {63}, journal = {SIAM Journal on Control and Optimization}, number = {1}, doi = {10.1137/23m1597332}, pages = {292 -- 320}, language = {en} }