@misc{StefanelliWachsmuthWachsmuth, author = {Stefanelli, Ulisse and Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Optimal control of a rate-independent evolution equation via viscous regularization}, series = {Discrete and Continuous Dynamical Systems - Series S (DCDS-S)}, volume = {10}, journal = {Discrete and Continuous Dynamical Systems - Series S (DCDS-S)}, number = {6}, issn = {1937-1632}, doi = {10.3934/dcdss.2017076}, pages = {1467 -- 1485}, abstract = {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.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints}, series = {Control and Cybernetics}, volume = {40}, journal = {Control and Cybernetics}, number = {4}, issn = {0324-8569}, pages = {1125 -- 1158}, abstract = {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.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Gerd and Wachsmuth, Daniel}, title = {Convergence and Regularization Results for Optimal Control Problems with Sparsity Functional}, series = {ESAIM: Control, Optimisation and Calculus of Variations}, volume = {17}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, number = {3}, issn = {1292-8119}, doi = {10.1051/cocv/2010027}, pages = {858 -- 886}, abstract = {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.}, language = {en} } @misc{GeigerWachsmuthWachsmuth, author = {Geiger, Tobias and Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Optimal control of ODEs with state suprema}, series = {Mathematical Control and Related Fields}, volume = {11}, journal = {Mathematical Control and Related Fields}, number = {3}, issn = {2156-8472}, doi = {10.3934/mcrf.2021012}, pages = {555 -- 578}, abstract = {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.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Necessary conditions for convergence rates of regularizations of optimal control problems}, series = {System Modeling and Optimization : 25th IFIP TC 7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers}, journal = {System Modeling and Optimization : 25th IFIP TC 7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers}, pages = {145 -- 154}, abstract = {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.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {How not to discretize the control}, series = {Special Issue:Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker-Vereinigung (DMV)}, volume = {16}, journal = {Special Issue:Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker-Vereinigung (DMV)}, number = {1}, address = {Braunschweig}, doi = {10.1002/pamm.201610385}, pages = {793 -- 794}, abstract = {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.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces}, series = {Journal of Convex Analysis}, volume = {30}, journal = {Journal of Convex Analysis}, number = {4}, pages = {1319 -- 1328}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 11}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Second-order conditions for non-uniformly convex integrands: quadratic growth in L¹}, series = {Journal of Nonsmooth Analysis and Optimizaton}, volume = {3}, journal = {Journal of Nonsmooth Analysis and Optimizaton}, issn = {2700-7448}, doi = {10.46298/jnsao-2022-8733}, pages = {1 -- 36}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Second-order conditions for non-uniformly convex integrands: quadratic growth in L^1}, series = {arXiv}, journal = {arXiv}, pages = {40}, abstract = {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.}, language = {en} } @misc{CasasWachsmuthWachsmuth, author = {Casas, Eduardo and Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Sufficient Second-Order Conditions for Bang-Bang Control Problems}, series = {SIAM Journal on Control and Optimization (SICON)}, volume = {55}, journal = {SIAM Journal on Control and Optimization (SICON)}, number = {5}, issn = {1095-7138}, doi = {10.1137/16M1099674}, pages = {3066 -- 3090}, abstract = {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)\$.}, language = {en} } @misc{CasasWachsmuthWachsmuth, author = {Casas, Eduardo and Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Second-Order Analysis and Numerical Approximation for Bang-Bang Bilinear Control Problems}, series = {SIAM Journal on Control and Optimization (SICON)}, volume = {56}, journal = {SIAM Journal on Control and Optimization (SICON)}, number = {6}, issn = {0363-0129}, doi = {10.1137/17M1139953}, pages = {4203 -- 4227}, abstract = {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.}, language = {en} } @misc{WachsmuthHerrmannHerzogetal., author = {Wachsmuth, Gerd and Herrmann, Marc and Herzog, Roland and Schmidt, Stephan and Vidal-N{\´u}{\~n}ez, Jos{\´e}}, title = {Discrete Total Variation with Finite Elements and Applications to Imaging}, series = {Journal of Mathematical Imaging and Vision : (JMIV)}, volume = {61}, journal = {Journal of Mathematical Imaging and Vision : (JMIV)}, number = {4}, issn = {0924-9907}, doi = {10.1007/s10851-018-0852-7}, pages = {411 -- 431}, abstract = {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.}, language = {en} } @misc{HarderWachsmuth, author = {Harder, Felix and Wachsmuth, Gerd}, title = {Optimality Conditions for a Class of Inverse Optimal Control Problems With Partial Differential Equations}, series = {Optimization: A Journal of Mathematical Programming and Operations Research}, journal = {Optimization: A Journal of Mathematical Programming and Operations Research}, issn = {0233-1934}, doi = {10.1080/02331934.2018.1495205}, pages = {29}, abstract = {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.}, 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{SchneiderWachsmuth, author = {Schneider, Christopher and Wachsmuth, Gerd}, title = {Regularization and discretization error estimates for optimal control of ODEs with group sparsity}, series = {ESAIM: Control, Optimisation and Calculus of Variations (COCV)}, volume = {24}, journal = {ESAIM: Control, Optimisation and Calculus of Variations (COCV)}, number = {2}, issn = {1292-8119}, doi = {10.1051/cocv/2017049}, pages = {811 -- 834}, abstract = {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.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Conforming Approximation of Convex Functions with the Finite Element Method}, series = {Numerische Mathematik}, volume = {137}, journal = {Numerische Mathematik}, number = {3}, issn = {0029-599X}, doi = {10.1007/s00211-017-0884-8}, pages = {741 -- 772}, abstract = {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.}, language = {en} } @misc{MehlitzWachsmuth, author = {Mehlitz, Patrick and Wachsmuth, Gerd}, title = {The Weak Sequential Closure of Decomposable Sets in Lebesgue Spaces and its Application to Variational Geometry}, series = {Set-Valued and Variational Analysis}, volume = {27}, journal = {Set-Valued and Variational Analysis}, number = {1}, issn = {1877-0533}, doi = {10.1007/s11228-017-0464-1}, pages = {265 -- 294}, abstract = {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.}, language = {en} } @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{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {On the Non-Polyhedricity of Sets with Upper and Lower Bounds in Dual Spaces}, series = {GAMM-Mitteilung}, volume = {40}, journal = {GAMM-Mitteilung}, number = {4}, issn = {0936-7195}, doi = {10.1002/gamm.201740005}, pages = {339 -- 350}, abstract = {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).}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Differentiability of implicit functions: Beyond the implicit function theorem}, series = {Journal of Mathematical Analysis and Applications (JMAA)}, volume = {414}, journal = {Journal of Mathematical Analysis and Applications (JMAA)}, number = {1}, issn = {0022-247X}, doi = {10.1016/j.jmaa.2014.01.007}, pages = {259 -- 272}, abstract = {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.}, language = {en} } @misc{WachsmuthLaetzerLeidich, author = {Wachsmuth, Gerd and L{\"a}tzer, Michael and Leidich, Erhard}, title = {Analytical computation of multiple interference fits under elasto-plastic deformations}, series = {Zeitschrift f{\"u}r angewandte Mathematik und Mechanik : (ZAMM)}, volume = {94}, journal = {Zeitschrift f{\"u}r angewandte Mathematik und Mechanik : (ZAMM)}, number = {12}, issn = {1521-4001}, doi = {10.1002/zamm.201300041}, pages = {1058 -- 1064}, abstract = {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.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {On LICQ and the uniqueness of Lagrange multipliers}, series = {Operations Research Letters}, volume = {41}, journal = {Operations Research Letters}, number = {1}, issn = {0167-6377}, doi = {10.1016/j.orl.2012.11.009}, pages = {78 -- 80}, abstract = {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.}, language = {en} } @misc{HerzogMeyerWachsmuth, author = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd}, title = {B- and Strong Stationarity for Optimal Control of Static Plasticity with Hardening}, series = {SIAM Journal on Optimization}, volume = {23}, journal = {SIAM Journal on Optimization}, number = {1}, issn = {1052-6234}, doi = {10.1137/110821147}, pages = {321 -- 352}, abstract = {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.}, language = {en} } @misc{HerzogMeyerWachsmuth, author = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd}, title = {C-Stationarity for Optimal Control of Static Plasticity with Linear Kinematic Hardening}, series = {SIAM Journal on Optimization}, volume = {50}, journal = {SIAM Journal on Optimization}, number = {5}, issn = {0363-0129}, doi = {10.1137/100809325}, pages = {3052 -- 3082}, abstract = {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.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Optimal Control of Quasi-static Plasticity with Linear Kinematic Hardening, Part I: Existence and Discretization in Time}, series = {SIAM Journal on Control and Optimization}, volume = {50}, journal = {SIAM Journal on Control and Optimization}, number = {5}, issn = {0363-0129}, doi = {10.1137/110839187}, pages = {2836 -- 2861}, abstract = {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.}, language = {en} } @misc{CasasHerzogWachsmuth, author = {Casas, Eduardo and Herzog, Roland and Wachsmuth, Gerd}, title = {Approximation of Sparse Controls in Semilinear Equations by Piecewise Linear Functions Numerische Mathematik}, series = {Numerische Mathematik}, volume = {122}, journal = {Numerische Mathematik}, number = {4}, issn = {0945-3245}, doi = {10.1007/s00211-012-0475-7}, pages = {645 -- 669}, abstract = {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.}, language = {en} } @misc{CasasHerzogWachsmuth, author = {Casas, Eduardo and Herzog, Roland and Wachsmuth, Gerd}, title = {Optimality Conditions and Error Analysis of Semilinear Elliptic Control Problems with L¹ Cost Functional}, series = {SIAM Journal on Optimization}, volume = {22}, journal = {SIAM Journal on Optimization}, number = {3}, issn = {1052-6234}, doi = {10.1137/110834366}, pages = {795 -- 820}, abstract = {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.}, language = {en} } @misc{HerzogMeyerWachsmuth, author = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd}, title = {Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions}, series = {Journal of Mathematical Analysis and Applications}, volume = {382}, journal = {Journal of Mathematical Analysis and Applications}, number = {2}, issn = {0022-247X}, doi = {10.1016/j.jmaa.2011.04.074}, pages = {802 -- 813}, abstract = {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.}, language = {en} } @incollection{MehlitzWachsmuth, author = {Mehlitz, Patrick and Wachsmuth, Gerd}, title = {Bilevel optimal control: existence results and stationarity conditions}, series = {Bilevel Optimization}, booktitle = {Bilevel Optimization}, publisher = {Springer Nature}, address = {Schweiz}, isbn = {978-3-030-52119-6}, doi = {10.1007/978-3-030-52119-6_16}, pages = {451 -- 484}, abstract = {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.}, language = {en} } @misc{HarderMehlitzWachsmuth, author = {Harder, Felix and Mehlitz, Patrick and Wachsmuth, Gerd}, title = {Reformulation of the M-stationarity conditions as a system of discontinuous equations and its solution by a semismooth Newton method}, series = {SIAM Journal on Optimization (SIOPT)}, volume = {31}, journal = {SIAM Journal on Optimization (SIOPT)}, number = {2}, issn = {1095-7189}, doi = {10.1137/20m1321413}, pages = {1459 -- 1488}, abstract = {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.}, language = {en} } @misc{HerzogMeyerWachsmuth, author = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd}, title = {Existence and regularity of the plastic multiplier in static and quasistatic plasticity}, series = {GAMM-Mitteilungen}, volume = {34}, journal = {GAMM-Mitteilungen}, number = {1}, issn = {1522-2608}, doi = {10.1002/gamm.201110006}, pages = {39 -- 44}, abstract = {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}, language = {en} } @misc{FrankeserHillerWachsmuthetal., author = {Frankeser, Sophia and Hiller, Sebastian and Wachsmuth, Gerd and Lutz, Josef}, title = {Using the on-state-Vbe,sat-voltage for temperature estimation of SiC-BJTs during normal operation}, series = {PCIM Europe : International Exhibition and Conference for Power Electronics, Intelligent Motion, Renewable Energy and Energy Management, Nuremberg, 19-21 May 2015 : proceedings}, journal = {PCIM Europe : International Exhibition and Conference for Power Electronics, Intelligent Motion, Renewable Energy and Energy Management, Nuremberg, 19-21 May 2015 : proceedings}, publisher = {VDE}, address = {Berlin ; Offenbach}, isbn = {978-3-8007-3924-0}, abstract = {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.}, language = {en} } @misc{NeitzelWachsmuth, author = {Neitzel, Ira and Wachsmuth, Gerd}, title = {First-order conditions for the optimal control of the obstacle problem with state constraints}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 35}, abstract = {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.}, language = {en} } @misc{LokutsievskiyWachsmuthZelikin, author = {Lokutsievskiy, Lev and Wachsmuth, Gerd and Zelikin, Mikhail}, title = {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}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 20}, abstract = {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.}, language = {en} } @misc{SchneiderWachsmuth, author = {Schneider, Ren{\´e} and Wachsmuth, Gerd}, title = {Achieving optimal convergence order for FEM in control constrained optimal control problems}, series = {Proceedings in Applied Mathematics and Mechanics (PAMM)}, volume = {15}, journal = {Proceedings in Applied Mathematics and Mechanics (PAMM)}, number = {1}, issn = {1617-7061}, doi = {10.1002/pamm.201510350}, pages = {731 -- 734}, abstract = {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.}, language = {en} } @misc{NeitzelWachsmuth, author = {Neitzel, Ira and Wachsmuth, Gerd}, title = {First-order conditions for the optimal control of the obstacle problem with state constraints}, series = {Pure and Applied Functional Analysis}, volume = {7}, journal = {Pure and Applied Functional Analysis}, number = {5}, issn = {2189-3764}, pages = {1881 -- 1911}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Non-existence of perturbed solutions under a second-order sufficient condition}, series = {Examples and Counterexamples}, volume = {4}, journal = {Examples and Counterexamples}, issn = {2666-657X}, doi = {10.1016/j.exco.2023.100122}, pages = {1 -- 3}, language = {en} } @misc{MarkoWachsmuth, author = {Marko, Jonas and Wachsmuth, Gerd}, title = {Integer optimal control problems with total variation regularization: Optimality conditions and fast solution of subproblems}, series = {ESAIM: Control, Optimisation and Calculus of Variations}, volume = {29}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, doi = {10.1051/cocv/2023065}, pages = {1 -- 30}, 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{KellerBartelsWachsmuth, author = {Keller, Hedwig and Bartels, S{\"o}ren and Wachsmuth, Gerd}, title = {Numerical Approximation of Optimal Convex and Rotationally Symmetric Shapes for an Eigenvalue Problem arising in Optimal Insulation}, series = {Computers \& Mathematics with Applications}, volume = {119}, journal = {Computers \& Mathematics with Applications}, issn = {1873-7668}, doi = {10.1016/j.camwa.2022.05.026}, pages = {327 -- 339}, language = {en} } @misc{AlphonseHintermuellerRautenbergetal., author = {Alphonse, Amal and Hinterm{\"u}ller, Michael and Rautenberg, Carlos N. and Wachsmuth, Gerd}, title = {Minimal and maximal solution maps of elliptic QVIs: penalisation, Lipschitz stability, differentiability and optimal control}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 33}, language = {en} } @misc{BorchardWachsmuth, author = {Borchard, Nicolas and Wachsmuth, Gerd}, title = {Second-order conditions for spatio-temporally sparse optimal control via second subderivatives}, series = {arXiv}, journal = {arXiv}, doi = {10.48550/arXiv.2311.14538}, pages = {1 -- 42}, language = {en} } @misc{BartelsKellerWachsmuth, author = {Bartels, S{\"o}ren and Keller, Hedwig and Wachsmuth, Gerd}, title = {Numerical Approximation of Optimal Convex Shapes in ℝ³}, series = {arXiv}, journal = {arXiv}, doi = {10.48550/arXiv.2311.13386}, pages = {1 -- 17}, 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{DempeFriedemannHarderetal., author = {Dempe, Stephan and Friedemann, Markus and Harder, Felix and Mehlitz, Patrick and Wachsmuth, Gerd}, title = {Bilevel optimal control: theory, algorithms, and applications}, series = {arXiv}, journal = {arXiv}, doi = {10.48550/arXiv.2305.19786}, pages = {1 -- 31}, language = {en} } @misc{LeidichLaetzerWachsmuth, author = {Leidich, Erhard and L{\"a}tzer, Michael and Wachsmuth, Gerd}, title = {Elastisch-plastisch beanspruchte Mehrfachpressverb{\"a}nde}, series = {4. VDI-Fachtagung Welle-Nabe-Verbindungen : Gestaltung, Fertigung, Anwendungen mit Fachausstellung, N{\"u}rtingen bei Stuttgart, 27. und 28. Oktober 2010}, journal = {4. VDI-Fachtagung Welle-Nabe-Verbindungen : Gestaltung, Fertigung, Anwendungen mit Fachausstellung, N{\"u}rtingen bei Stuttgart, 27. und 28. Oktober 2010}, publisher = {VDI-Verl.}, address = {D{\"u}sseldorf}, isbn = {978-3-18-092114-3}, pages = {335 -- 342}, language = {de} } @misc{FriedemannHarderWachsmuth, author = {Friedemann, Markus and Harder, Felix and Wachsmuth, Gerd}, title = {Finding global solutions of some inverse optimal control problems using penalization and semismooth Newton methods}, series = {Journal of Global Optimization}, volume = {86}, journal = {Journal of Global Optimization}, number = {4}, issn = {1573-2916}, doi = {10.1007/s10898-023-01288-7}, pages = {1025 -- 1061}, language = {en} } @misc{JiaKanzowMehlitzetal., author = {Jia, Xiaoxi and Kanzow, Christian and Mehlitz, Patrick and Wachsmuth, Gerd}, title = {An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints}, series = {Mathematical Programming}, volume = {199}, journal = {Mathematical Programming}, doi = {10.1007/s10107-022-01870-z}, pages = {1365 -- 1415}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Maximal monotone operators with non-maximal graphical limit}, series = {Examples and Counterexamples}, volume = {2}, journal = {Examples and Counterexamples}, issn = {2666-657X}, doi = {10.1016/j.exco.2022.100073}, language = {en} }