TY - GEN A1 - Harder, Felix A1 - Wachsmuth, Gerd T1 - Optimality Conditions for a Class of Inverse Optimal Control Problems With Partial Differential Equations T2 - Optimization: A Journal of Mathematical Programming and Operations Research N2 - We consider bilevel optimization problems which can be interpreted as inverse optimal control problems. The lower-level problem is an optimal control problem with a parametrized objective function. The upper-level problem is used to identify the parameters of the lower-level problem. Our main focus is the derivation of first-order necessary optimality conditions. We prove C-stationarity of local solutions of the inverse optimal control problem and give a counterexample to show that strong stationarity might be violated at a local minimizer. KW - Inverse optimal control problem KW - optimality condition KW - C-stationarity KW - strong stationarity Y1 - 2018 UR - https://www.tandfonline.com/doi/full/10.1080/02331934.2018.1495205 U6 - https://doi.org/10.1080/02331934.2018.1495205 SN - 0233-1934 SN - 1029-4945 ER - TY - GEN A1 - Wachsmuth, Gerd A1 - Harder, Felix T1 - The limiting normal cone of a complementarity set in Sobolev spaces T2 - Optimization N2 - 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. KW - Limiting normal cone KW - optimality condition KW - M-stationarity KW - obstacle problem Y1 - 2018 U6 - https://doi.org/10.1080/02331934.2018.1484467 SN - 0233-1934 SN - 1029-4945 VL - 67 IS - 10 SP - 1579 EP - 1603 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 - TY - GEN A1 - Christof, Constantin A1 - Wachsmuth, Gerd T1 - On the Non‐Polyhedricity of Sets with Upper and Lower Bounds in Dual Spaces T2 - GAMM-Mitteilung N2 - We demonstrate that the set L∞(X, [−1,1]) of all measurable functions over a Borel measure space (X, B, μ) with values in the unit interval is typically non‐polyhedric when interpreted as a subset of a dual space. Our findings contrast the classical result that subsets of Dirichlet spaces with pointwise upper and lower bounds are polyhedric. In particular, additional structural assumptions are unavoidable when the concept of polyhedricity is used to study the differentiability properties of solution maps to variational inequalities of the second kind in, e.g., the spaces H1/2(∂Ω)or H₀¹(Ω). KW - Polydedricity KW - Frictional Contact Problems KW - Variational Inequalities of the Second Kind KW - Directional Differentiability KW - Sensitivity Analysis Y1 - 2018 U6 - https://doi.org/10.1002/gamm.201740005 SN - 0936-7195 SN - 1522-2608 VL - 40 IS - 4 SP - 339 EP - 350 ER - TY - GEN A1 - Harder, Felix A1 - Wachsmuth, Gerd T1 - Comparison of Optimality Systems for the Optimal Control of the Obstacle Problem T2 - GAMM-Mitteilung N2 - 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. KW - obstacle problem KW - complementarity constraints KW - weak stationarity KW - M-stationarity KW - strong stationarity Y1 - 2018 U6 - https://doi.org/10.1002/gamm.201740004 SN - 0936-7195 SN - 1522-2608 VL - 40 IS - 4 SP - 312 EP - 338 ER - TY - GEN A1 - Schneider, Christopher A1 - Wachsmuth, Gerd T1 - Regularization and discretization error estimates for optimal control of ODEs with group sparsity T2 - ESAIM: Control, Optimisation and Calculus of Variations (COCV) N2 - It is well known that optimal control problems with L¹-control costs produce sparse solutions, i.e., the optimal control is zero on whole intervals. In this paper, we study a general class of convex linear-quadratic optimal control problems with a sparsity functional that promotes a so-called group sparsity structure of the optimal controls. In this case, the components of the control function take the value of zero on parts of the time interval, simultaneously. These problems are both theoretically interesting and practically relevant. After obtaining results about the structure of the optimal controls, we derive stability estimates for the solution of the problem w.r.t. perturbations and L²-regularization. These results are consequently applied to prove convergence of the Euler discretization. Finally, the usefulness of our approach is demonstrated by solving an illustrative example using a semismooth Newton method. KW - Optimal control KW - group sparsity KW - directional sparsity KW - bang-bang principle KW - stability analysis KW - discretization error estimates Y1 - 2018 U6 - https://doi.org/10.1051/cocv/2017049 SN - 1292-8119 SN - 1262-3377 VL - 24 IS - 2 SP - 811 EP - 834 ER - TY - GEN A1 - Wachsmuth, Gerd T1 - Conforming Approximation of Convex Functions with the Finite Element Method T2 - Numerische Mathematik N2 - We consider the interior approximation of convex functions with convex finite element functions. The main motivation for this study is the investigation of a novel discretization of optimization problems with convexity constraints by the finite element method. Under a mild assumption on the family of meshes, we show that the conforming approximation is convergent if the finite elements are at least piecewise quadratic. We further provide similar results under additional constraints on the function values or on the gradient. The theoretical findings are illustrated by numerical examples. Y1 - 2017 U6 - https://doi.org/10.1007/s00211-017-0884-8 SN - 0029-599X SN - 0945-3245 VL - 137 IS - 3 SP - 741 EP - 772 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 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - The Weak Sequential Closure of Decomposable Sets in Lebesgue Spaces and its Application to Variational Geometry T2 - Set-Valued and Variational Analysis N2 - We provide a precise characterization of the weak sequential closure of nonempty, closed, decomposable sets in Lebesgue spaces. Therefore, we have to distinguish between the purely atomic and the nonatomic regime. In the latter case, we get a convexification effect which is related to Lyapunov’s convexity theorem, and in the former case, the weak sequential closure equals the strong closure. The characterization of the weak sequential closure is utilized to compute the limiting normal cone to nonempty, closed, decomposable sets in Lebesgue spaces. Finally, we give an example for the possible nonclosedness of the limiting normal cone in this setting. KW - Decomposable set KW - Lebesgue spaces KW - Limiting normal cone KW - Measurability KW - Weak sequential closure Y1 - 2019 U6 - https://doi.org/10.1007/s11228-017-0464-1 SN - 1877-0533 SN - 1877-0541 VL - 27 IS - 1 SP - 265 EP - 294 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 - 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 - Wachsmuth, Gerd T1 - Strong stationarity for optimization problems with complementarity constraints in absence of polyhedricity T2 - Set-Valued and Variational Analysis N2 - We consider mathematical programs with complementarity constraints in Banach spaces. In particular, we focus on the situation that the complementarity constraint is defined by a non-polyhedric cone K. We demonstrate how strong stationarity conditions can be obtained in an abstract setting. These conditions and their verification can be made more precise in the case that Z is a Hilbert space and if the projection onto K is directionally differentiable with a derivative as given in Haraux (Journal of the Mathematical Society of Japan 29(4), 615–631, 1977, Theorem 1). Finally, we apply the theory to optimization problems with semidefinite and second-order-cone complementarity constraints. We obtain that local minimizers are strongly stationary under a variant of the linear-independence constraint qualification, and these are novel results. KW - Mathematical programs with complementarity constraints KW - Strong stationarity KW - Conic programming KW - Semidefinite cone KW - Second-order cone KW - Polyhedricity Y1 - 2017 U6 - https://doi.org/10.1007/s11228-016-0370-y SN - 1877-0533 SN - 1877-0541 VL - 25 IS - 1 SP - 133 EP - 175 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 - 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 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - Weak and strong stationarity in generalized bilevel programming and bilevel optimal control T2 - Optimization N2 - In this article, we consider a general bilevel programming problem in reflexive Banach spaces with a convex lower level problem. In order to derive necessary optimality conditions for the bilevel problem, it is transferred to a mathematical program with complementarity constraints (MPCC). We introduce a notion of weak stationarity and exploit the concept of strong stationarity for MPCCs in reflexive Banach spaces, recently developed by the second author, and we apply these concepts to the reformulated bilevel programming problem. Constraint qualifications are presented, which ensure that local optimal solutions satisfy the weak and strong stationarity conditions. Finally, we discuss a certain bilevel optimal control problem by means of the developed theory. Its weak and strong stationarity conditions of Pontryagin-type and some controllability assumptions ensuring strong stationarity of any local optimal solution are presented. KW - Bilevel programming KW - programming in Banach spaces KW - mathematical program with complementarity constraints KW - stationarity KW - bilevel optimal control Y1 - 2016 U6 - https://doi.org/10.1080/02331934.2015.1122007 SN - 0233-1934 SN - 1029-4945 VL - 65 IS - 5 SP - 907 EP - 935 ER - TY - GEN A1 - Casas, Eduardo A1 - Herzog, Roland A1 - Wachsmuth, Gerd T1 - Analysis of Spatio-Temporally Sparse Optimal Control Problems of Semilinear Parabolic Equations T2 - ESAIM: Control, Optimisation and Calculus of Variations (COCV) N2 - Optimal control problems with semilinear parabolic state equations are considered. The objective features one out of three different terms promoting various spatio-temporal sparsity patterns of the control variable. For each problem, first-order necessary optimality conditions, as well as second-order necessary and sufficient optimality conditions are proved. The analysis includes the case in which the objective does not contain the squared norm of the control. KW - Optimal control KW - directional sparsity KW - second-order optimality conditions KW - semilinear parabolic equations Y1 - 2017 U6 - https://doi.org/10.1051/cocv/2015048 SN - 1292-8119 SN - 1262-3377 VL - 23 IS - 1 SP - 263 EP - 295 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 - Wachsmuth, Gerd T1 - Optimal Control of Quasistatic Plasticity with Linear Kinematic Hardening III: Optimality Conditions T2 - Zeitschrift für Analysis und ihre Anwendungen N2 - In this paper we consider an optimal control problem governed by a rate-independent variational inequality arising in quasistatic plasticity with linear kinematic hardening. Since the solution operator of a variational inequality is not diff� erentiable, the Karush-Kuhn-Tucker system is not a necessary optimality condition. We show a system of weakly stationary type by passing to the limit with the optimality system of a regularized and time-discretized problem. KW - Complementarity condition KW - quasistatic plasticity KW - time-dependent variational inequality KW - mathematical program with complementarity constraints KW - evolution variational inequality KW - rate-independent Y1 - 2016 U6 - https://doi.org/10.4171/ZAA/1556 SN - 0232-2064 SN - 1661-4354 VL - 35 IS - 1 SP - 81 EP - 118 ER - TY - GEN A1 - Wachsmuth, Gerd T1 - Optimal Control of Quasistatic Plasticity with Linear Kinematic Hardening II: Regularization and Differentiability T2 - Zeitschrift für Analysis und ihre Anwendungen (ZAA) N2 - We consider an optimal control problem governed by an evolution variational inequality arising in quasistatic plasticity with linear kinematic hardening. A regularization of the time-discrete problem is derived. The regularized forward problem can be interpreted as system of coupled quasilinear PDEs whose principal parts depend on the gradient of the state. We show the Fréchet differentiability of the solution map of this quasilinear system. As a consequence, we obtain a first order necessary optimality system. Moreover, we address certain convergence properties of the regularization. KW - Complementarity condition KW - quasistatic plasticity KW - time-dependent variational inequality KW - mathematical program with complementarity constraints KW - evolution variational inequality KW - rate-independent Y1 - 2015 U6 - https://doi.org/10.4171/ZAA/1546 SN - 0232-2064 SN - 1661-4534 VL - 34 IS - 4 SP - 391 EP - 418 ER -