TY - GEN A1 - Schneider, René A1 - Wachsmuth, Gerd T1 - A-posteriori error estimation for control-constrained, linear-quadratic optimal control problems T2 - SIAM Journal on Numerical Analysis N2 - We derive a posteriori error estimates for control-constrained, linear-quadratic optimal control problems. The error is measured in a norm which is motivated by the objective. Our abstract error estimator is separated into three contributions: the error in the variational inequality (i.e., in the optimality condition for the control) and the errors in the state and adjoint equation. Hence, one can use well-established estimators for the differential equations. We show that the abstract error estimator is reliable and efficient if the utilized estimators for the differential equations have these properties. We apply the error estimator to two distributed optimal control problems with distributed and boundary observation, respectively. Numerical examples exhibit a good error reduction if we use the local error contributions for an adaptive mesh refinement. KW - adaptive finite elements KW - a posteriori error analysis KW - optimal control KW - control constraints Y1 - 2016 U6 - https://doi.org/10.1137/15M1020460 SN - 1095-7170 SN - 0036-1429 VL - 54 IS - 2 SP - 1169 EP - 1192 ER - TY - GEN A1 - Wachsmuth, Gerd T1 - Towards M-stationarity for optimal control of the obstacle problem with control constraints T2 - SIAM Journal on Control and Optimization N2 - We consider an optimal control problem, whose state is given as the solution of the obstacle problem. The controls are not assumed to be dense in H⁻¹(\Ω). Hence, local minimizers may not be strongly stationary. By a nonsmooth regularization technique similar to the virtual control regularization, we prove a system of C-stationarity using only minimal regularity requirements. We show that even a system of M-stationarity is satisfied under the assumption that the regularized adjoint states converge in capacity. We also give a counterexample, showing that this additional assumption might be crucial. KW - M-stationarity conditions KW - obstalce problem KW - control constraints Y1 - 2016 U6 - https://doi.org/10.1137/140980582 SN - 1095-7138 SN - 0363-0129 VL - 54 IS - 2 SP - 964 EP - 986 ER - TY - GEN A1 - Wachsmuth, Gerd T1 - Pointwise Constraints in Vector-Valued Sobolev Spaces. With Applications in Optimal Control T2 - Applied Mathematics & Optimization N2 - We consider a set C with pointwise constraints in a vector-valued Sobolev space. We characterize its tangent and normal cone. Under the additional assumption that the pointwise constraints are affine and satisfy the linear independence constraint qualification, we show that the set C is polyhedric. The results are applied to the optimal control of a string in a polyhedral tube. KW - Tangent cone KW - Normal cone KW - Polyhedricity KW - Vector-valued function KW - Vector-valued measure Y1 - 2016 U6 - https://doi.org/10.1007/s00245-016-9381-1 SN - 0095-4616 SN - 1432-0606 VL - 77 IS - 3 SP - 463 EP - 497 ER - TY - GEN A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - The limiting normal cone to pointwise defined sets in Lebesgue spaces T2 - Set-Valued and Variational Analysis (SVAA) N2 - We consider subsets of Lebesgue spaces which are defined by pointwise constraints. We provide formulas for corresponding variational objects (tangent and normal cones). Our main result shows that the limiting normal cone is always dense in the Clarke normal cone and contains the convex hull of the pointwise limiting normal cone. A crucial assumption for this result is that the underlying measure is non-atomic, and this is satisfied in many important applications (Lebesgue measure on subsets of Rͩ or the surface measure on hypersurfaces in Rͩ). Finally, we apply our findings to an optimization problem with complementarity constraints in Lebesgue spaces. KW - Decomposable set KW - Lebesgue spaces KW - Limiting normal cone KW - Mathematical program with complementarity constraint KW - Measurability Y1 - 2018 U6 - https://doi.org/10.1007/s11228-016-0393-4 SN - 1877-0533 SN - 1877-0541 VL - 26 IS - 3 SP - 449 EP - 467 ER - TY - GEN A1 - Rösch, Arnd A1 - Wachsmuth, Gerd T1 - Mass lumping for the optimal control of elliptic partial differential equations T2 - SIAM Journal on Numerical Analysis (SINUM) N2 - The finite element discretization of a control constrained elliptic optimal control problem is studied. Control and state are discretized by higher order finite elements. The inequality constraints are only posed in the Lagrange points. The computational effort is significantly reduced by a new mass lumping strategy. The main contribution is the derivation of new a priori error estimates up to order h⁴ on locally refined meshes. Moreover, we propose a new algorithmic strategy to obtain such highly accurate results. The theoretical findings are illustrated by numerical examples. KW - Optimal control KW - control constraint KW - higher order finite elements KW - mass lumping KW - a priori error estimates Y1 - 2017 U6 - https://doi.org/10.1137/16M1074473 SN - 0036-1429 SN - 1095-7170 VL - 55 IS - 3 SP - 1412 EP - 1436 ER - TY - GEN A1 - González Merino, Bernardo A1 - Jahn, Thomas A1 - Polyanskii, Alexander A1 - Wachsmuth, Gerd T1 - Hunting for Reduced Polytopes T2 - Discrete & Computational Geometry N2 - We show that there exist reduced polytopes in three-dimensional Euclidean space. This partially answers the question posed by Lassak (Israel J Math 70(3):365–379, 1990) on the existence of reduced polytopes in d-dimensional Euclidean space for d≥3. KW - Minimum width KW - Polytope KW - Reducedness Y1 - 2018 U6 - https://doi.org/10.1007/s00454-018-9982-3 SN - 0179-5376 SN - 1432-0444 VL - 60 IS - 3 SP - 801 EP - 808 ER - TY - GEN A1 - Wachsmuth, Gerd T1 - A guided tour of polyhedric sets: basic properties, new results on intersections and applications T2 - Journal of Convex Analysis N2 - The aim of this contribution is twofold. On the one hand, we give some new results concerning polyhedric sets. In particular, we show that sets with pointwise lower and upper bound are polyhedric in many important function spaces. Moreover, we show that the intersection of such a set with finitely many hyperplanes and half-spaces is polyhedric. We also provide counterexamples demonstrating that the intersection of polyhedric sets may fail to be polyhedric. On the other hand, we gather all important results from the literature concerning polyhedric sets in order to give a complete picture of the current knowledge. In particular, we illustrate the applications of polyhedricity. KW - Polyhedricity KW - polyhedric set KW - directional differentiability KW - projection KW - vector lattice KW - strong stationarity KW - second-order conditions Y1 - 2019 SN - 0944-6532 VL - 26 IS - 1 SP - 153 EP - 188 ER - TY - GEN A1 - Christof, Constantin A1 - Wachsmuth, Gerd T1 - No-Gap Second-Order Conditions via a Directional Curvature Functional T2 - SIAM Journal on Optimization N2 - This paper is concerned with necessary and sufficient second-order conditions for finite-dimensional and infinite-dimensional constrained optimization problems. Using a suitably defined directional curvature functional for the admissible set, we derive no-gap second-order optimality conditions in an abstract functional analytic setting. Our theory not only covers those cases where the classical assumptions of polyhedricity or second-order regularity are satisfied but also allows to study problems in the absence of these requirements. As a tangible example, we consider no-gap second-order conditions for bang-bang optimal control problems. KW - second-order condition KW - bang-bang control KW - polyhedricity KW - second-order regularity KW - no-gap optimality condition Y1 - 2018 U6 - https://doi.org/10.1137/17M1140418 SN - 1052-6234 SN - 1095-7189 VL - 28 IS - 3 SP - 2097 EP - 2130 ER -