TY - GEN A1 - Stefanelli, Ulisse A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Optimal control of a rate-independent evolution equation via viscous regularization T2 - Discrete and Continuous Dynamical Systems - Series S (DCDS-S) N2 - We study the optimal control of a rate-independent system that is driven by a convex quadratic energy. Since the associated solution mapping is non-smooth, the analysis of such control problems is challenging. In order to derive optimality conditions, we study the regularization of the problem via a smoothing of the dissipation potential and via the addition of some viscosity. The resulting regularized optimal control problem is analyzed. By driving the regularization parameter to zero, we obtain a necessary optimality condition for the original, non-smooth problem. KW - Rate-independent system KW - optimal control KW - necessary optimality conditions Y1 - 2017 U6 - https://doi.org/10.3934/dcdss.2017076 SN - 1937-1632 SN - 1937-1179 VL - 10 IS - 6 SP - 1467 EP - 1485 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints T2 - Control and Cybernetics N2 - In this article we study the regularization of optimization problems by Tikhonov regularization. The optimization problems are subject to pointwise inequality constraints in L²(Ω). We derive a-priori regularization error estimates if the regularization parameter as well as the noise level tend to zero. We rely on an assumption that is a combination of a source condition and of a structural assumption on the active sets. Moreover, we introduce a strategy to choose the regularization parameter in dependence of the noise level. We prove convergence of this parameter choice rule with optimal order. KW - source condition KW - discrepancy principle KW - non-smooth optimization KW - convex constraints KW - sparsity KW - regularization error estimates Y1 - 2011 SN - 0324-8569 VL - 40 IS - 4 SP - 1125 EP - 1158 ER - TY - GEN A1 - Wachsmuth, Gerd A1 - Wachsmuth, Daniel T1 - Convergence and Regularization Results for Optimal Control Problems with Sparsity Functional T2 - ESAIM: Control, Optimisation and Calculus of Variations N2 - Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered. The non-smoothness arises from a L1-norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newton method. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical experiments. KW - Non-smooth optimization KW - sparsity KW - regularization error estimates KW - finite elements KW - discretization error estimates Y1 - 2011 U6 - https://doi.org/10.1051/cocv/2010027 SN - 1292-8119 SN - 1262-3377 VL - 17 IS - 3 SP - 858 EP - 886 ER - TY - GEN A1 - Geiger, Tobias A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Optimal control of ODEs with state suprema T2 - Mathematical Control and Related Fields N2 - We consider the optimal control of a differential equation that involves the suprema of the state over some part of the history. In many applications, this non-smooth functional dependence is crucial for the successful modeling of real-world phenomena. We prove the existence of solutions and show that related problems may not possess optimal controls. Due to the non-smoothness in the state equation, we cannot obtain optimality conditions via standard theory. Therefore, we regularize the problem via a LogIntExp functional which generalizes the well-known LogSumExp. By passing to the limit with the regularization, we obtain an optimality system for the original problem. The theory is illustrated by some numerical experiments. KW - Functional differential equations KW - differential equations with state suprema KW - optimality conditions KW - maximum principle KW - LogIntExp Y1 - 2021 U6 - https://doi.org/10.3934/mcrf.2021012 SN - 2156-8472 SN - 2156-8499 VL - 11 IS - 3 SP - 555 EP - 578 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Necessary conditions for convergence rates of regularizations of optimal control problems T2 - System Modeling and Optimization : 25th IFIP TC 7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers N2 - We investigate the Tikhonov regularization of control constrained optimal control problems. We use a specialized source condition in combination with a condition on the active sets. In the case of high convergence rates, these conditions are necessary and sufficient. KW - optimal control problem, inequality constraints, Tikhonov regularization, source condition Y1 - 2012 UR - https://link.springer.com/chapter/10.1007/978-3-642-36062-6_15 SP - 145 EP - 154 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - How not to discretize the control T2 - Special Issue:Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker‐Vereinigung (DMV) N2 - In this short note, we address the discretization of optimal control problems with higher order polynomials. We develop a necessary and sufficient condition to ensure that weak limits of discrete feasible controls are feasible for the original problem. We show by means of a simple counterexample that a naive discretization by higher order polynomials can lead to non-feasible limits of sequences of discrete solutions. Y1 - 2016 UR - https://arxiv.org/abs/1603.07099 U6 - https://doi.org/10.1002/pamm.201610385 VL - 16 IS - 1 SP - 793 EP - 794 CY - Braunschweig ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces T2 - Journal of Convex Analysis Y1 - 2023 UR - https://www.heldermann.de/JCA/JCA30/JCA304/jca30059.htm VL - 30 IS - 4 SP - 1319 EP - 1328 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces T2 - arXiv Y1 - 2022 UR - https://arxiv.org/abs/2204.00282 SP - 1 EP - 11 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Second-order conditions for non-uniformly convex integrands: quadratic growth in L¹ T2 - Journal of Nonsmooth Analysis and Optimizaton Y1 - 2022 U6 - https://doi.org/10.46298/jnsao-2022-8733 SN - 2700-7448 VL - 3 SP - 1 EP - 36 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Second-order conditions for non-uniformly convex integrands: quadratic growth in L^1 T2 - arXiv N2 - We study no-gap second-order optimality conditions for a non-uniformly convex and non-smooth integral functional. The integral functional is extended to the space of measures. The obtained second-order derivatives contain integrals on lower-dimensional manifolds. The proofs utilize the convex pre-conjugate, which is an integral functional on the space of continuous functions. Application to non-smooth optimal control problems are given. KW - Second-order optimality conditions KW - twice epi-differentiability KW - bang-bang control KW - sparse control Y1 - 2021 UR - https://arxiv.org/pdf/2111.10238.pdf ER - TY - GEN A1 - Casas, Eduardo A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Sufficient Second-Order Conditions for Bang-Bang Control Problems T2 - SIAM Journal on Control and Optimization (SICON) N2 - We provide sufficient optimality conditions for optimal control problems with bang-bang controls. Building on a structural assumption on the adjoint state, we additionally need a weak second-order condition. This second-order condition is formulated with functions from an extended critical cone, and it is equivalent to a formulation posed on measures supported on the set where the adjoint state vanishes. If our sufficient optimality condition is satisfied, we obtain a local quadratic growth condition in $L^1(\Omega)$. KW - bang-bang control KW - second-order conditions KW - sufficient optimality KW - critical cone Y1 - 2017 U6 - https://doi.org/10.1137/16M1099674 SN - 1095-7138 SN - 0363-0129 VL - 55 IS - 5 SP - 3066 EP - 3090 ER - TY - GEN A1 - Casas, Eduardo A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Second-Order Analysis and Numerical Approximation for Bang-Bang Bilinear Control Problems T2 - SIAM Journal on Control and Optimization (SICON) N2 - We consider bilinear optimal control problems whose objective functionals do not depend on the controls. Hence, bang-bang solutions will appear. We investigate sufficient second-order conditions for bang-bang controls, which guarantee local quadratic growth of the objective functional in $L^1$. In addition, we prove that for controls that are not bang-bang, no such growth can be expected. Finally, we study the finite-element discretization and prove error estimates of bang-bang controls in $L^1$-norms. KW - bang-bang control KW - bilinear controls KW - second-order conditions KW - sufficient optimality conditions KW - error analysis Y1 - 2018 U6 - https://doi.org/10.1137/17M1139953 SN - 0363-0129 SN - 1095-7138 VL - 56 IS - 6 SP - 4203 EP - 4227 ER - TY - GEN A1 - Wachsmuth, Gerd A1 - Herrmann, Marc A1 - Herzog, Roland A1 - Schmidt, Stephan A1 - Vidal-Núñez, José T1 - Discrete Total Variation with Finite Elements and Applications to Imaging T2 - Journal of Mathematical Imaging and Vision : (JMIV) N2 - The total variation (TV)-seminorm is considered for piecewise polynomial, globally discontinuous (DG) and continuous (CG) finite element functions on simplicial meshes. A novel, discrete variant (DTV) based on a nodal quadrature formula is defined. DTV has favorable properties, compared to the original TV-seminorm for finite element functions. These include a convenient dual representation in terms of the supremum over the space of Raviart–Thomas finite element functions, subject to a set of simple constraints. It can therefore be shown that a variety of algorithms for classical image reconstruction problems, including TV-L2 and TV-L1, can be implemented in low and higher-order finite element spaces with the same efficiency as their counterparts originally developed for images on Cartesian grids. KW - Discrete total variaton KW - Dual problem KW - Image reconstruction KW - Numerical algorithms Y1 - 2019 U6 - https://doi.org/10.1007/s10851-018-0852-7 SN - 0924-9907 SN - 1573-7683 VL - 61 IS - 4 SP - 411 EP - 431 ER - 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 - 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 - 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 - 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 - 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 - Wachsmuth, Gerd T1 - Differentiability of implicit functions: Beyond the implicit function theorem T2 - Journal of Mathematical Analysis and Applications (JMAA) N2 - The implicit function theorem (IFT) can be used to deduce the differentiability of an implicit mapping S : u → y given by the equation e(y,u) = 0. However, the IFT is not applicable when different norms are necessary for the differentiation of e w.r.t. y and the invertibility of the partial derivative eᵧ(y,u). We prove theorems ensuring the (twice) differentiability of the mapping S which can be applied in this case. We highlight the application of our results to quasilinear partial differential equations whose principal part depends nonlinearly on the gradient of the state ∇y. KW - Implicit function theorem KW - Differentiability KW - Quasilinear partial differential equations Y1 - 2014 U6 - https://doi.org/10.1016/j.jmaa.2014.01.007 SN - 0022-247X VL - 414 IS - 1 SP - 259 EP - 272 ER - TY - GEN A1 - Wachsmuth, Gerd A1 - Lätzer, Michael A1 - Leidich, Erhard T1 - Analytical computation of multiple interference fits under elasto‐plastic deformations T2 - Zeitschrift für angewandte Mathematik und Mechanik : (ZAMM) N2 - Multiple interference fits represent high friction‐stressed shaft‐hub connections. High surface pressures together with thin‐walled intermediate parts lead to high stresses and showing strong tendencies to plastic deformations. Cylindrical interference fits and their elasto‐plastic behaviour have been analysed in the literature. However, in the case of multiple interference fits, these methods are not applicable. In this paper, we derive a new calculation method for elasto‐plastically deformed multiple interference fits with hardening material. The analytical method is validated by the comparison of the determined stresses and displacements of several test cases with the solutions of corresponding axisymmetric finite element models. KW - interference fit KW - plastic deformation Y1 - 2013 U6 - https://doi.org/10.1002/zamm.201300041 SN - 1521-4001 VL - 94 IS - 12 SP - 1058 EP - 1064 ER - TY - GEN A1 - Wachsmuth, Gerd T1 - On LICQ and the uniqueness of Lagrange multipliers T2 - Operations Research Letters N2 - Kyparisis proved in 1985 that a strict version of the Mangasarian–Fromovitz constraint qualification (MFCQ) is equivalent to the uniqueness of Lagrange multipliers. However, the definition of this strict version of MFCQ requires the existence of a Lagrange multiplier and is not a constraint qualification (CQ) itself. In this note we show that LICQ is the weakest CQ which ensures the (existence and) uniqueness of Lagrange multipliers. We also recall the relations between other CQs and properties of the set of Lagrange multipliers. KW - Lagrange multiplier KW - Constraint qualification Y1 - 2013 U6 - https://doi.org/10.1016/j.orl.2012.11.009 SN - 0167-6377 VL - 41 IS - 1 SP - 78 EP - 80 ER - TY - GEN A1 - Herzog, Roland A1 - Meyer, Christian A1 - Wachsmuth, Gerd T1 - B- and Strong Stationarity for Optimal Control of Static Plasticity with Hardening T2 - SIAM Journal on Optimization N2 - Optimal control problems for the variational inequality of static elastoplasticity with linear kinematic hardening are considered. The control-to-state map is shown to be weakly directionally differentiable, and local optimal controls are proved to verify an optimality system of B-stationary type. For a modified problem, local minimizers are shown to even satisfy an optimality system of strongly stationary type. KW - mathematical programs with complementarity constraints in function space KW - variational inequalities of first kind KW - elastoplasticity KW - Bouligand and strong stationarity Y1 - 2013 U6 - https://doi.org/10.1137/110821147 SN - 1052-6234 SN - 1095-7189 VL - 23 IS - 1 SP - 321 EP - 352 ER - TY - GEN A1 - Herzog, Roland A1 - Meyer, Christian A1 - Wachsmuth, Gerd T1 - C-Stationarity for Optimal Control of Static Plasticity with Linear Kinematic Hardening T2 - SIAM Journal on Optimization N2 - An optimal control problem is considered for the variational inequality representing the stress-based (dual) formulation of static elastoplasticity. The linear kinematic hardening model and the von Mises yield condition are used. Existence and uniqueness of the plastic multiplier is rigorously proved, which allows for the reformulation of the forward system using a complementarity condition. In order to derive necessary optimality conditions, a family of regularized optimal control problems is analyzed, wherein the static plasticity problems are replaced by their viscoplastic approximations. By passing to the limit in the optimality conditions for the regularized problems, necessary optimality conditions of C-stationarity type are obtained. KW - mathematical programs with complementarity constraints KW - static plasticity KW - optimal control KW - optimality conditions Y1 - 2012 U6 - https://doi.org/10.1137/100809325 SN - 0363-0129 SN - 1095-7138 VL - 50 IS - 5 SP - 3052 EP - 3082 ER - TY - GEN A1 - Wachsmuth, Gerd T1 - Optimal Control of Quasi-static Plasticity with Linear Kinematic Hardening, Part I: Existence and Discretization in Time T2 - SIAM Journal on Control and Optimization N2 - In this paper we consider an optimal control problem governed by a time-dependent variational inequality arising in quasi-static plasticity with linear kinematic hardening. We address certain continuity properties of the forward operator, which imply the existence of an optimal control. Moreover, a discretization in time is derived and we show that every local minimizer of the continuous problem can be approximated by minimizers of modified, time-discrete problems. KW - complementarity condition KW - quasistatic plasticity KW - time-dependent variational inequality KW - mathematical program with complementarity constraints KW - evolution variational inequality KW - rate-dependent Y1 - 2012 U6 - https://doi.org/10.1137/110839187 SN - 0363-0129 SN - 1095-7138 VL - 50 IS - 5 SP - 2836 EP - 2861 ER - TY - GEN A1 - Casas, Eduardo A1 - Herzog, Roland A1 - Wachsmuth, Gerd T1 - Approximation of Sparse Controls in Semilinear Equations by Piecewise Linear Functions Numerische Mathematik T2 - Numerische Mathematik N2 - Semilinear elliptic optimal control problems involving the L¹ norm of the control in the objective are considered. A priori finite element error estimates for piecewise linear discretizations for the control and the state are proved. These are obtained by a new technique based on an appropriate discretization of the objective function. Numerical experiments confirm the convergence rates. Y1 - 2012 U6 - https://doi.org/10.1007/s00211-012-0475-7 SN - 0945-3245 SN - 0029-599X VL - 122 IS - 4 SP - 645 EP - 669 ER - TY - GEN A1 - Casas, Eduardo A1 - Herzog, Roland A1 - Wachsmuth, Gerd T1 - Optimality Conditions and Error Analysis of Semilinear Elliptic Control Problems with L¹ Cost Functional T2 - SIAM Journal on Optimization N2 - Semilinear elliptic optimal control problems involving the L¹ norm of the control in the objective are considered. Necessary and sufficient second-order optimality conditions are derived. A priori finite element error estimates for piecewise constant discretizations for the control and piecewise linear discretizations of the state are shown. Error estimates for the variational discretization of the problem in the sense of [M. Hinze, Comput. Optim. Appl., 30 (2005), pp. 45–61] are also obtained. Numerical experiments confirm the convergence rates. KW - optimal control of partial differential equations KW - nondifferentiable objective KW - sparse controls KW - finite element discretization KW - a priori error estimates Y1 - 2012 U6 - https://doi.org/10.1137/110834366 SN - 1052-6234 SN - 1095-7189 VL - 22 IS - 3 SP - 795 EP - 820 ER - TY - GEN A1 - Herzog, Roland A1 - Meyer, Christian A1 - Wachsmuth, Gerd T1 - Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions T2 - Journal of Mathematical Analysis and Applications N2 - Equations of linear and nonlinear infinitesimal elasticity with mixed boundary conditionsare considered. The bounded domain is assumed to have a Lipschitz boundary and to satisfy additional regularity assumptions. W^(1,p) regularity for the displacements and L^p regularity for the stresses are proved for some p>2. KW - Nonlinear elasticity KW - Integrability KW - Mixed boundary conditions KW - Discontinuous coefficients Y1 - 2011 U6 - https://doi.org/10.1016/j.jmaa.2011.04.074 SN - 0022-247X VL - 382 IS - 2 SP - 802 EP - 813 ER - TY - CHAP A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - Bilevel optimal control: existence results and stationarity conditions T2 - Bilevel Optimization N2 - The mathematical modeling of numerous real-world applications results in hierarchical optimization problems with two decision makers where at least one of them has to solve an optimal control problem of ordinary or partial differential equations. Such models are referred to as bilevel optimal control problems. Here, we first review some different features of bilevel optimal control including important applications, existence results, solution approaches, and optimality conditions. Afterwards, we focus on a specific problem class where parameters appearing in the objective functional of an optimal control problem of partial differential equations have to be reconstructed. After verifying the existence of solutions, necessary optimality conditions are derived by exploiting the optimal value function of the underlying parametric optimal control problem in the context of a relaxation approach. KW - Bilevel optimal control KW - Existence results KW - Inverse optimal control KW - Stationarity conditions Y1 - 2020 SN - 978-3-030-52119-6 SN - 978-3-030-52118-9 U6 - https://doi.org/10.1007/978-3-030-52119-6_16 SP - 451 EP - 484 PB - Springer Nature CY - Schweiz ER - TY - GEN A1 - Harder, Felix A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - Reformulation of the M-stationarity conditions as a system of discontinuous equations and its solution by a semismooth Newton method T2 - SIAM Journal on Optimization (SIOPT) N2 - We show that the Mordukhovich-stationarity system associated with a mathematical program with complementarity constraints (MPCC) can be equivalently written as a system of discontinuous equations which can be tackled with a semismooth Newton method. It will be demonstrated that the resulting algorithm can be interpreted as an active set strategy for MPCCs. Local fast convergence of the method is guaranteed under validity of an MPCC-tailored version of LICQ and a suitable strong second-order condition. In case of linear-quadratic MPCCs, the LICQ-type constraint qualification can be replaced by a weaker condition which depends on the underlying multipliers. We discuss a suitable globalization strategy for our method. Some numerical results are presented in order to illustrate our theoretical findings. KW - active set method KW - mathematical program with complementarity constraints KW - M-stationarity KW - nonlinear M-stationarity function KW - semismooth Newton method Y1 - 2021 U6 - https://doi.org/10.1137/20m1321413 SN - 1095-7189 SN - 1052-6234 VL - 31 IS - 2 SP - 1459 EP - 1488 ER - TY - GEN A1 - Herzog, Roland A1 - Meyer, Christian A1 - Wachsmuth, Gerd T1 - Existence and regularity of the plastic multiplier in static and quasistatic plasticity T2 - GAMM-Mitteilungen N2 - Existence of the plastic multiplier with L1 spatial regularity for quasistatic and static plasticity is proved for arbitrary continuous and convex yield functions and linear hardening laws. L2 regularity is shown in the particular cases of kinematic hardening, or combined kinematic and isotropic hardening KW - Plastic Multiplier KW - Static Plasticity KW - Quasistatic Plasticity KW - Variational Inequalities Y1 - 2011 U6 - https://doi.org/10.1002/gamm.201110006 SN - 1522-2608 VL - 34 IS - 1 SP - 39 EP - 44 ER - TY - GEN A1 - Frankeser, Sophia A1 - Hiller, Sebastian A1 - Wachsmuth, Gerd A1 - Lutz, Josef T1 - Using the on-state-Vbe,sat-voltage for temperature estimation of SiC-BJTs during normal operation T2 - PCIM Europe : International Exhibition and Conference for Power Electronics, Intelligent Motion, Renewable Energy and Energy Management, Nuremberg, 19-21 May 2015 : proceedings N2 - Temperature measurement of power semiconductor devices in real-life application is a desired goal to implement health monitoring methods like thermal impedance spectroscopy. During on-state, the base-emitter voltage of current driven SiC bipolar transistors shows dependency on the collector current but also on the temperature. It is planned to measure the virtual junction temperature of a SiC-BJT through electric sensitive parameters during onstate. Y1 - 2015 SN - 978-3-8007-3924-0 PB - VDE CY - Berlin ; Offenbach ER - TY - GEN A1 - Neitzel, Ira A1 - Wachsmuth, Gerd T1 - First-order conditions for the optimal control of the obstacle problem with state constraints T2 - arXiv N2 - We consider an optimal control problem in which the state is governed by an unilateral obstacle problem (with obstacle from below) and restricted by a pointwise state constraint (from above). In the presence of control constraints, we prove, via regularization of the state constraints, that a system of C-stationarity is necessary for optimality. In the absence of control constraints, we show that local minimizers are even strongly stationary by a careful discussion of the primal first-order conditions of B-stationary type. KW - Obstacle problem KW - state constraints KW - C-stationarity KW - strong stationarity Y1 - 2020 UR - https://arxiv.org/abs/2012.15324 SP - 1 EP - 35 ER - TY - GEN A1 - Lokutsievskiy, Lev A1 - Wachsmuth, Gerd A1 - Zelikin, Mikhail T1 - Non-optimality of conical parts for Newton's problem of minimal resistance in the class of convex bodies and the limiting case of infinite height T2 - arXiv N2 - We consider Newton's problem of minimal resistance, in particular we address the problem arising in the limit if the height goes to infinity. We establish existence of solutions and lack radial symmetry of solutions. Moreover, we show that certain conical parts contained in the boundary of a convex body inhibit the optimality in the classical Newton's problem with finite height. This result is applied to certain bodies considered in the literature, which are conjectured to be optimal for the classical Newton's problem, and we show that they are not. KW - Newton’s problem of minimal resistance KW - Conical parts KW - Convex bodies Y1 - 2021 UR - https://arxiv.org/pdf/2009.12128 SP - 1 EP - 20 ER - TY - GEN A1 - Schneider, René A1 - Wachsmuth, Gerd T1 - Achieving optimal convergence order for FEM in control constrained optimal control problems T2 - Proceedings in Applied Mathematics and Mechanics (PAMM) N2 - We propose a new heuristic approach to overcome convergence order restrictions implied by the low regularity of the optimal control due to the activity interface of control constraints. Aligning the mesh with the interface yields an improved approximation of the control. Utility of the approach is demonstrated by numerical experiments. Y1 - 2015 U6 - https://doi.org/10.1002/pamm.201510350 SN - 1617-7061 VL - 15 IS - 1 SP - 731 EP - 734 ER - TY - GEN A1 - Neitzel, Ira A1 - Wachsmuth, Gerd T1 - First-order conditions for the optimal control of the obstacle problem with state constraints T2 - Pure and Applied Functional Analysis Y1 - 2022 UR - http://yokohamapublishers.jp/online2/oppafa/vol7/p1881.html SN - 2189-3764 SN - 2189-3756 VL - 7 IS - 5 SP - 1881 EP - 1911 ER - TY - GEN A1 - Wachsmuth, Gerd T1 - Non-existence of perturbed solutions under a second-order sufficient condition T2 - Examples and Counterexamples Y1 - 2023 U6 - https://doi.org/10.1016/j.exco.2023.100122 SN - 2666-657X VL - 4 SP - 1 EP - 3 ER - TY - GEN A1 - Marko, Jonas A1 - Wachsmuth, Gerd T1 - Integer optimal control problems with total variation regularization: Optimality conditions and fast solution of subproblems T2 - ESAIM: Control, Optimisation and Calculus of Variations Y1 - 2023 U6 - https://doi.org/10.1051/cocv/2023065 VL - 29 SP - 1 EP - 30 ER - TY - GEN A1 - Christof, Constantin A1 - Wachsmuth, Gerd T1 - Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control T2 - SIAM Journal on Control and Optimization Y1 - 2023 U6 - https://doi.org/10.1137/21m1467365 SN - 1095-7138 SN - 0887-4603 VL - 61 IS - 3 SP - 1162 EP - 1186 ER - TY - GEN A1 - Keller, Hedwig A1 - Bartels, Sören A1 - Wachsmuth, Gerd T1 - Numerical Approximation of Optimal Convex and Rotationally Symmetric Shapes for an Eigenvalue Problem arising in Optimal Insulation T2 - Computers & Mathematics with Applications Y1 - 2022 U6 - https://doi.org/10.1016/j.camwa.2022.05.026 SN - 1873-7668 SN - 0898-1221 VL - 119 SP - 327 EP - 339 ER - TY - GEN A1 - Alphonse, Amal A1 - Hintermüller, Michael A1 - Rautenberg, Carlos N. A1 - Wachsmuth, Gerd T1 - Minimal and maximal solution maps of elliptic QVIs: penalisation, Lipschitz stability, differentiability and optimal control T2 - arXiv Y1 - 2023 UR - https://arxiv.org/abs/2312.13879 SP - 1 EP - 33 ER - TY - GEN A1 - Borchard, Nicolas A1 - Wachsmuth, Gerd T1 - Second-order conditions for spatio-temporally sparse optimal control via second subderivatives T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2311.14538 SP - 1 EP - 42 ER - TY - GEN A1 - Bartels, Sören A1 - Keller, Hedwig A1 - Wachsmuth, Gerd T1 - Numerical Approximation of Optimal Convex Shapes in ℝ³ T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2311.13386 SP - 1 EP - 17 ER - TY - GEN A1 - Christof, Constantin A1 - Wachsmuth, Gerd T1 - Energy Space Newton Differentiability for Solution Maps of Unilateral and Bilateral Obstacle Problems T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2308.15289 SP - 1 EP - 26 ER - TY - GEN A1 - Dempe, Stephan A1 - Friedemann, Markus A1 - Harder, Felix A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - Bilevel optimal control: theory, algorithms, and applications T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2305.19786 SP - 1 EP - 31 ER - TY - GEN A1 - Leidich, Erhard A1 - Lätzer, Michael A1 - Wachsmuth, Gerd T1 - Elastisch-plastisch beanspruchte Mehrfachpressverbände T2 - 4. VDI-Fachtagung Welle-Nabe-Verbindungen : Gestaltung, Fertigung, Anwendungen mit Fachausstellung, Nürtingen bei Stuttgart, 27. und 28. Oktober 2010 Y1 - 2010 SN - 978-3-18-092114-3 SP - 335 EP - 342 PB - VDI-Verl. CY - Düsseldorf ER - TY - GEN A1 - Friedemann, Markus A1 - Harder, Felix A1 - Wachsmuth, Gerd T1 - Finding global solutions of some inverse optimal control problems using penalization and semismooth Newton methods T2 - Journal of Global Optimization Y1 - 2023 U6 - https://doi.org/10.1007/s10898-023-01288-7 SN - 1573-2916 SN - 0925-5001 VL - 86 IS - 4 SP - 1025 EP - 1061 ER - TY - GEN A1 - Jia, Xiaoxi A1 - Kanzow, Christian A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints T2 - Mathematical Programming Y1 - 2023 U6 - https://doi.org/10.1007/s10107-022-01870-z VL - 199 SP - 1365 EP - 1415 ER - TY - GEN A1 - Wachsmuth, Gerd T1 - Maximal monotone operators with non-maximal graphical limit T2 - Examples and Counterexamples Y1 - 2022 U6 - https://doi.org/10.1016/j.exco.2022.100073 SN - 2666-657X VL - 2 ER -