@misc{WachsmuthHarder, author = {Wachsmuth, Gerd and Harder, Felix}, title = {The limiting normal cone of a complementarity set in Sobolev spaces}, series = {Optimization}, volume = {67}, journal = {Optimization}, number = {10}, issn = {0233-1934}, doi = {10.1080/02331934.2018.1484467}, pages = {1579 -- 1603}, abstract = {We investigate the limiting normal cone of the complementarity set associated with non-negative functions in the Sobolev space. By using results from homogenization theory, we provide lower estimates for this limiting normal cone. These estimates are unpleasantly large.}, language = {en} } @misc{HarderWachsmuth, author = {Harder, Felix and Wachsmuth, Gerd}, title = {Comparison of Optimality Systems for the Optimal Control of the Obstacle Problem}, series = {GAMM-Mitteilung}, volume = {40}, journal = {GAMM-Mitteilung}, number = {4}, issn = {0936-7195}, doi = {10.1002/gamm.201740004}, pages = {312 -- 338}, abstract = {We consider stationarity systems for the optimal control of the obstacle problem. The focus is on the comparison of several different systems which are provided in the literature. We obtain some novel results concerning the relations between these stationarity concepts.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Strong Stationarity for Optimal Control of the Obstacle Problem with Control Constraints}, series = {SIAM Journal on Optimization :(SIOPT)}, volume = {24}, journal = {SIAM Journal on Optimization :(SIOPT)}, number = {4}, issn = {1052-6234}, doi = {10.1137/130925827}, pages = {1914 -- 1932}, abstract = {We consider the distributed optimal control of the obstacle problem with control constraints. Since Mignot proved in 1976 the necessity of a system which is equivalent to strong stationarity, it has been an open problem whether such a system is still necessary in the presence of control constraints. Using moderate regularity of the optimal control and an assumption on the control bounds (which is implied by \$u_a < 0 \le u_b\$ quasi-everywhere in \$\Omega\$ in the case of an upper obstacle \$y \le \psi\$), we can answer this question in the affirmative. We also present counterexamples showing that strong stationarity may not hold if \$u_a < 0\$ or \$0 \le u_b\$ are violated.}, language = {en} } @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 = {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} }