@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} } @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}, series = {Calculus of Variations and Partial Differential Equations}, volume = {61}, journal = {Calculus of Variations and Partial Differential Equations}, number = {1}, issn = {1432-0835}, doi = {10.1007/s00526-021-02118-y}, language = {en} } @misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacle-Type Quasi-Variational Inequalities}, series = {SIAM Journal on Control and Optimization}, volume = {60}, journal = {SIAM Journal on Control and Optimization}, number = {6}, issn = {1095-7138}, doi = {10.1137/21m1419635}, pages = {3430 -- 3456}, language = {en} } @misc{MehlitzWachsmuth, author = {Mehlitz, Patrick and Wachsmuth, Gerd}, title = {Subdifferentiation of nonconvex sparsity-promoting functionals on Lebesgue spaces}, series = {SIAM Journal on Control and Optimization}, volume = {60}, journal = {SIAM Journal on Control and Optimization}, number = {3}, issn = {1095-7138}, doi = {10.1137/21m1435173}, pages = {1819 -- 1839}, language = {en} } @misc{HarderWachsmuth, author = {Harder, Felix and Wachsmuth, Gerd}, title = {M-stationarity for a class of MPCCs in Lebesgue spaces}, series = {Journal of Mathematical Analysis and Applications}, volume = {512}, journal = {Journal of Mathematical Analysis and Applications}, number = {2}, pages = {1 -- 28}, language = {en} } @misc{KanzowKraemerMehlitzetal., author = {Kanzow, Christian and Kr{\"a}mer, Fabius and Mehlitz, Patrick and Wachsmuth, Gerd and Werner, Frank}, title = {A nonsmooth augmented Lagrangian method and its application to Poisson denoising and sparse control}, series = {arXiv}, journal = {arXiv}, doi = {10.48550/arXiv.2304.06434}, pages = {1 -- 36}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Non-existence of perturbed solutions under a second-order sufficient condition}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 4}, language = {en} } @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 = {arXiv}, journal = {arXiv}, pages = {1 -- 45}, language = {en} } @misc{MehlitzWachsmuth, author = {Mehlitz, Patrick and Wachsmuth, Gerd}, title = {Subdifferentiation of nonconvex sparsity-promoting functionals on Lebesgue spaces}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 25}, abstract = {Sparsity-promoting terms are incorporated into the objective functions of optimal control problems in order to ensure that optimal controls vanish on large parts of the underlying domain. Typical candidates for those terms are integral functions on Lebesgue spaces based on the ℓp-metric for p∈[0,1) which are nonconvex as well as non-Lipschitz and, thus, variationally challenging. In this paper, we derive exact formulas for the Fr{\´e}chet, limiting, and singular subdifferential of these functionals. These generalized derivatives can be used for the derivation of necessary optimality conditions for optimal control problems comprising such sparsity-promoting terms.}, language = {en} } @misc{EtlingHerzogLoayzaetal., author = {Etling, Tommy and Herzog, Roland and Loayza, Estefania and Wachsmuth, Gerd}, title = {First and Second Order Shape Optimization Based on Restricted Mesh Deformations}, series = {SIAM Journal on Scientific Computing}, volume = {42}, journal = {SIAM Journal on Scientific Computing}, number = {2}, issn = {1095-7197}, doi = {10.1137/19M1241465}, pages = {A1200 -- A1225}, abstract = {We consider shape optimization problems subject to elliptic partial differential equations. In the context of the finite element method, the geometry to be optimized is represented by the computational mesh, and the optimization proceeds by repeatedly updating the mesh node positions. It is well known that such a procedure eventually may lead to a deterioration of mesh quality, or even an invalidation of the mesh, when interior nodes penetrate neighboring cells. We examine this phenomenon, which can be traced back to the ineptness of the discretized objective when considered over the space of mesh node positions. As a remedy, we propose a restriction in the admissible mesh deformations, inspired by the Hadamard structure theorem. First and second order methods are considered in this setting. Numerical results show that mesh degeneracy can be overcome, avoiding the need for remeshing or other strategies. FEniCS code for the proposed methods is available on GitHub.}, language = {en} } @misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {On Second-Order Optimality Conditionsfor Optimal Control Problems Governedby the Obstacle Problem}, series = {Optimization}, journal = {Optimization}, issn = {1029-4945}, doi = {10.1080/02331934.2020.1778686}, pages = {41}, abstract = {This paper is concerned with second-order optimality conditions for Tikhonov regularized optimal control problems governed by the obstacle problem. Using a simple observation that allows to characterize the structure of optimal controls on the active set, we derive various conditions that guarantee the local/global optimality of first-order stationary points and/or the local/global quadratic growth of the reduced objective function. Our analysis extends and refines existing results from the literature and also covers those situations where the problem at hand involves additional box-constraints on the control. As a byproduct, our approach shows in particular that Tikhonov regularized optimal control problems for the obstacle problem can be reformulated as state-constrained optimal control problems for the Poisson equation and that problems involving a subharmonic obstacle and a convex objective function are uniquely solvable. The paper concludes with three counterexamples which illustrate that rather peculiar effects can occur in the analysis of second-order optimality conditions for optimal control problems governed by the obstacle problem and that necessary second-order conditions for such problems may be hard to derive.}, language = {en} } @misc{BoergensKanzowMehlitzetal., author = {B{\"o}rgens, Eike and Kanzow, Christian and Mehlitz, Patrick and Wachsmuth, Gerd}, title = {New Constraint Qualifications for Optimization Problems in Banach Spaces Based on Asymptotic KKT Conditions}, series = {SIAM Journal on Optimization}, volume = {30}, journal = {SIAM Journal on Optimization}, number = {4}, issn = {1095-7189}, doi = {10.1137/19M1306804}, pages = {2956 -- 2982}, abstract = {Optimization theory in Banach spaces suffers from a lack of available constraint qualifications. There exist very few constraint qualifications, and these are often violated even in simple applications. This is very much in contrast to finite-dimensional nonlinear programs, where a large number of constraint qualifications is known. Since these constraint qualifications are usually defined using the set of active inequality constraints, it is difficult to extend them to the infinite-dimensional setting. One exception is a recently introduced sequential constraint qualification based on asymptotic KKT conditions. This paper shows that this so-called asymptotic KKT regularity allows suitable extensions to the Banach space setting in order to obtain new constraint qualifications. The relation of these new constraint qualifications to existing ones is discussed in detail. Their usefulness is also shown by several examples as well as an algorithmic application to the class of augmented Lagrangian methods.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {No-Gap Second-Order Conditions under n-Polyhedric Constraints and Finitely Many Nonlinear Constraints}, series = {Journal of Convex Analysis}, volume = {27}, journal = {Journal of Convex Analysis}, number = {2}, issn = {2363-6394}, pages = {733 -- 751}, abstract = {We consider an optimization problem subject to an abstract constraint and finitely many nonlinear constraints. Using the recently introduced concept of n-polyhedricity, we are able to provide second-order optimality conditions under weak regularity assumptions. In particular, we prove necessary optimality conditions of first and second order under the constraint qualification of Robinson, Zowe and Kurcyusz. Similarly, sufficient optimality conditions are stated. The gap between both conditions is as small as possible.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Elliptic quasi-variational inequalities under a smallness assumption: uniqueness, differential stability and optimal control}, series = {Calculus of Variations and Partial Differential Equations}, volume = {59}, journal = {Calculus of Variations and Partial Differential Equations}, number = {2}, issn = {1432-0835}, doi = {10.1007/s00526-020-01743-3}, pages = {15}, abstract = {We consider a quasi-variational inequality governed by a moving set. We employ the assumption that the movement of the set has a small Lipschitz constant. Under this requirement, we show that the quasi-variational inequality has a unique solution which depends Lipschitz-continuously on the source term. If the data of the problem is (directionally) differentiable, the solution map is directionally differentiable as well. We also study the optimal control of the quasi-variational inequality and provide necessary optimality conditions of strongly stationary type.}, language = {en} } @misc{BartelsWachsmuth, author = {Bartels, S{\"o}ren and Wachsmuth, Gerd}, title = {Numerical Approximation of Optimal Convex Shapes}, series = {SIAM Journal on Scientific Computing}, volume = {42}, journal = {SIAM Journal on Scientific Computing}, number = {2}, issn = {1095-7197}, doi = {10.1137/19M1256853}, pages = {A1226 -- A1244}, abstract = {This article investigates the numerical approximation of shape optimization problems with PDE constraint on classes of convex domains. The convexity constraint provides a compactness property which implies well posedness of the problem. Moreover, we prove the convergence of discretizations in two-dimensional situations. A numerical algorithm is devised that iteratively solves the discrete formulation. Numerical experiments show that optimal convex shapes are generally nonsmooth and that three-dimensional problems require an appropriate relaxation of the convexity condition.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability}, series = {Journal of Nonsmooth Analysis and Optimizaton}, volume = {3}, journal = {Journal of Nonsmooth Analysis and Optimizaton}, issn = {2700-7448}, doi = {10.46298/jnsao-2022-8537}, pages = {1 -- 22}, language = {en} } @misc{DominguezCorellaWachsmuth, author = {Dom{\´i}nguez Corella, Alberto and Wachsmuth, Gerd}, title = {Stability and genericity of bang-bang controls in affine problems}, series = {arXiv}, journal = {arXiv}, doi = {10.48550/arXiv.2307.05418}, pages = {1 -- 25}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Slater conditions without interior points for programs in Lebesgue spaces with pointwise bounds and finitely many constraints}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 20}, 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 = {arXiv}, journal = {arXiv}, pages = {1 -- 36}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Engel, Sebastian; Kunisch, Karl: Optimal control of the linear wave equation by time-depending BV-controls: a semi-smooth Newton approach}, series = {Mathematical Control and Related Fields}, volume = {10}, journal = {Mathematical Control and Related Fields}, number = {3}, issn = {2156-8499}, pages = {591 -- 622}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Banz, Lothar; Hinterm{\"u}ller, Michael; Schr{\"o}der, Andreas: A posteriori error control for distributed elliptic optimal control problems with control constraints discretized by \$hp\$-finite elements.}, series = {Comput. Math. Appl. 80 (2020), no. 11, 2433-2450}, journal = {Comput. Math. Appl. 80 (2020), no. 11, 2433-2450}, publisher = {American Mathematical Society}, address = {Providence}, language = {en} } @misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 24}, abstract = {We prove that solution operators of elliptic obstacle-type variational inequalities (or, more generally, locally Lipschitz continuous functions possessing certain pointwise-a.e. convexity properties) are Newton differentiable when considered as maps between suitable Lebesgue spaces and equipped with the strong-weak Bouligand differential as a generalized set-valued derivative. It is shown that this Newton differentiability allows to solve optimal control problems with H1-cost terms and one-sided pointwise control constraints by means of a semismooth Newton method. The superlinear convergence of the resulting algorithm is proved in the infinite-dimensional setting and its mesh independence is demonstrated in numerical experiments. We expect that the findings of this paper are also helpful for the design of numerical solution procedures for quasi-variational inequalities and the optimal control of obstacle-type variational problems.}, language = {en} } @misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacle-Type Quasi-Variational Inequalities}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 26}, abstract = {This paper is concerned with the sensitivity analysis of a class of parameterized fixed-point problems that arise in the context of obstacle-type quasi-variational inequalities. We prove that, if the operators in the considered fixed-point equation satisfy a positive superhomogeneity condition, then the maximal and minimal element of the solution set of the problem depend locally Lipschitz continuously on the involved parameters. We further show that, if certain concavity conditions hold, then the maximal solution mapping is Hadamard directionally differentiable and its directional derivatives are precisely the minimal solutions of suitably defined linearized fixed-point equations. In contrast to prior results, our analysis requires neither a Dirichlet space structure, nor restrictive assumptions on the mapping behavior and regularity of the involved operators, nor sign conditions on the directions that are considered in the directional derivatives. Our approach further covers the elliptic and parabolic setting simultaneously and also yields Hadamard directional differentiability results in situations in which the solution set of the fixed-point equation is a continuum and a characterization of directional derivatives via linearized auxiliary problems is provably impossible. To illustrate that our results can be used to study interesting problems arising in practice, we apply them to establish the Hadamard directional differentiability of the solution operator of a nonlinear elliptic quasi-variational inequality, which emerges in impulse control and in which the obstacle mapping is obtained by taking essential infima over certain parts of the underlying domain, and of the solution mapping of a parabolic quasi-variational inequality, which involves boundary controls and in which the state-to-obstacle relationship is described by a partial differential equation.}, 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 = {arXiv}, journal = {arXiv}, pages = {1 -- 49}, abstract = {This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints are nonconvex and possibly complicated, but allow for a fast computation of projections onto this nonconvex set. Typical problem classes which satisfy this requirement are optimization problems with disjunctive constraints (like complementarity or cardinality constraints) as well as optimization problems over sets of matrices which have to satisfy additional rank constraints. The key idea behind our method is to keep these complicated constraints explicitly in the constraints and to penalize only the remaining constraints by an augmented Lagrangian function. The resulting subproblems are then solved with the aid of a problem-tailored nonmonotone projected gradient method. The corresponding convergence theory allows for an inexact solution of these subproblems. Nevertheless, the overall algorithm computes so-called Mordukhovich-stationary points of the original problem under a mild asymptotic regularity condition, which is generally weaker than most of the respective available problem-tailored constraint qualifications. Extensive numerical experiments addressing complementarity- and cardinality-constrained optimization problems as well as a semidefinite reformulation of Maxcut problems visualize the power of our approach.}, language = {en} } @misc{HarderWachsmuth, author = {Harder, Felix and Wachsmuth, Gerd}, title = {M-stationarity for a class of MPCCs in Lebesgue spaces}, series = {arXiv}, journal = {arXiv}, abstract = {We show that an optimality condition of M-stationarity type holds for minimizers of a class of mathematical programs with complementarity constraints (MPCCs) in Lebesgue spaces. We apply these results also to local minimizers of an inverse optimal control problem (which is an instance of an infinite-dimensional bilevel optimization problem). The multipliers for the M-stationarity system can be constructed via convex combinations of various multipliers to auxiliary, linear problems. However, proving the existence of the multipliers to these auxiliary problems is difficult and only possible in some situations.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Maximal monotone operators with non-maximal graphical limit}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 4}, abstract = {We present a counterexample showing that the graphical limit of maximally monotone operators might not be maximally monotone. We also characterize the directional differentiability of the resolvent of an operator B in terms of existence and maximal monotonicity of the proto-derivative of B.}, language = {en} } @incollection{HerzogMeyerWachsmuth, author = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd}, title = {Optimal Control of Elastoplastic Processes: Analysis, Algorithms, Numerical Analysis and Applications}, series = {Trends in PDE Constrained Optimization}, booktitle = {Trends in PDE Constrained Optimization}, editor = {Leugering, G{\"u}nter}, edition = {1. Auflage}, publisher = {Birkh{\"a}user}, address = {Cham}, isbn = {978-3-319-05082-9}, doi = {10.1007/978-3-319-05083-6_4}, pages = {27 -- 41}, 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. The forward system is reformulated such that it involves the plastic multiplier and a complementarity condition. In order to derive necessary optimality conditions, a family of regularized optimal control problems is analyzed. C-stationarity type conditions are obtained by passing to the limit with the regularization. Numerical results are presented.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 27}, abstract = {We consider a generalized equation governed by a strongly monotone and Lipschitz single-valued mapping and a maximally monotone set-valued mapping in a Hilbert space. We are interested in the sensitivity of solutions w.r.t. perturbations of both mappings. We demonstrate that the directional differentiability of the solution map can be verified by using the directional differentiability of the single-valued operator and of the resolvent of the set-valued mapping. The result is applied to quasi-generalized equations in which we have an additional dependence of the solution within the set-valued part of the equation.}, 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}, pages = {34}, abstract = {We are interested in the optimization of convex domains under a PDE constraint. Due to the difficulties of approximating convex domains in R^3, the restriction to rotationally symmetric domains is used to reduce shape optimization problems to a two-dimensional setting. For the optimization of an eigenvalue arising in a problem of optimal insulation, the existence of an optimal domain is proven. An algorithm is proposed that can be applied to general shape optimization problems under the geometric constraints of convexity and rotational symmetry. The approximated optimal domains for the eigenvalue problem in optimal insulation are discussed.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {The numerical solution of Newton's problem of least resistance}, series = {Mathematical Programming}, volume = {147}, journal = {Mathematical Programming}, number = {1-2}, issn = {0025-5610}, doi = {10.1007/s10107-014-0756-2}, pages = {331 -- 350}, abstract = {In this paper we consider Newton's problem of finding a convex body of least resistance. This problem could equivalently be written as a variational problem over concave functions in R2. We propose two different methods for solving it numerically. First, we discretize this problem by writing the concave solution function as a infimum over a finite number of affine functions. The discretized problem could be solved by standard optimization software efficiently. Second, we conjecture that the optimal body has a certain structure. We exploit this structure and obtain a variational problem in R1. Deriving its Euler-Lagrange equation yields a program with two unknowns, which can be solved quickly.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Strong Stationarity for Optimal Control of the Obstacle Problem with Control Constraints}, series = {SIAM Journal on Optimization :(SIOPT)}, volume = {24}, journal = {SIAM Journal on Optimization :(SIOPT)}, number = {4}, issn = {1052-6234}, doi = {10.1137/130925827}, pages = {1914 -- 1932}, abstract = {We consider the distributed optimal control of the obstacle problem with control constraints. Since Mignot proved in 1976 the necessity of a system which is equivalent to strong stationarity, it has been an open problem whether such a system is still necessary in the presence of control constraints. Using moderate regularity of the optimal control and an assumption on the control bounds (which is implied by \$u_a < 0 \le u_b\$ quasi-everywhere in \$\Omega\$ in the case of an upper obstacle \$y \le \psi\$), we can answer this question in the affirmative. We also present counterexamples showing that strong stationarity may not hold if \$u_a < 0\$ or \$0 \le u_b\$ are violated.}, language = {en} } @misc{RaulsWachsmuth, author = {Rauls, Anne-Therese and Wachsmuth, Gerd}, title = {Generalized Derivatives for the Solution Operator of the Obstacle Problem}, series = {Set-Valued and Variational Analysis}, volume = {28}, journal = {Set-Valued and Variational Analysis}, number = {2}, issn = {1877-0533}, doi = {10.1007/s11228-019-0506-y}, pages = {259 -- 285}, abstract = {We characterize generalized derivatives of the solution operator of the obstacle problem. This precise characterization requires the usage of the theory of so-called capacitary measures and the associated solution operators of relaxed Dirichlet problems. The generalized derivatives can be used to obtain a novel necessary optimality condition for the optimal control of the obstacle problem with control constraints. A comparison shows that this system is stronger than the known system of C-stationarity.}, language = {en} } @misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {Differential Sensitivity Analysis of Variational Inequalities with Locally Lipschitz Continuous Solution Operators}, series = {Applied Mathematics \& Optimization}, volume = {81}, journal = {Applied Mathematics \& Optimization}, number = {1}, issn = {0095-4616}, doi = {10.1007/s00245-018-09553-y}, pages = {23 -- 62}, abstract = {This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). Our method of proof is fully elementary. Moreover, our technique allows us to also study those cases where the variational inequality at hand is not uniquely solvable and where directional differentiability can only be obtained w.r.t. the weak or the weak-star topology of the underlying space. As tangible examples, we consider a variational inequality arising in elastoplasticity, the projection onto prox-regular sets, and a bang-bang optimal control problem.}, language = {en} } @misc{DempeHarderMehlitzetal., author = {Dempe, Stephan and Harder, Felix and Mehlitz, Patrick and Wachsmuth, Gerd}, title = {Solving inverse optimal control problems via value functions to global optimality}, series = {Journal of Global Optimization}, volume = {74}, journal = {Journal of Global Optimization}, number = {2}, issn = {0925-5001}, doi = {10.1007/s10898-019-00758-1}, pages = {297 -- 325}, abstract = {In this paper, we show how a special class of inverse optimal control problems of elliptic partial differential equations can be solved globally. Using the optimal value function of the underlying parametric optimal control problem, we transfer the overall hierarchical optimization problem into a nonconvex single-level one. Unfortunately, standard regularity conditions like Robinson's CQ are violated at all the feasible points of this surrogate problem. It is, however, shown that locally optimal solutions of the problem solve a Clarke-stationarity-type system. Moreover, we relax the feasible set of the surrogate problem iteratively by approximating the lower level optimal value function from above by piecewise affine functions. This allows us to compute globally optimal solutions of the original inverse optimal control problem. The global convergence of the resulting algorithm is shown theoretically and illustrated by means of a numerical example.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Optimal Control of Quasistatic Plasticity with Linear Kinematic Hardening II: Regularization and Differentiability}, series = {Zeitschrift f{\"u}r Analysis und ihre Anwendungen (ZAA)}, volume = {34}, journal = {Zeitschrift f{\"u}r Analysis und ihre Anwendungen (ZAA)}, number = {4}, issn = {0232-2064}, doi = {10.4171/ZAA/1546}, pages = {391 -- 418}, abstract = {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 Fr{\´e}chet 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.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Mathematical Programs with Complementarity Constraints in Banach Spaces}, series = {Journal of Optimization Theory and Applications (JOTA)}, volume = {166}, journal = {Journal of Optimization Theory and Applications (JOTA)}, number = {2}, issn = {0022-3239}, doi = {10.1007/s10957-014-0695-3}, pages = {480 -- 507}, abstract = {We consider optimization problems in Banach spaces involving a complementarity constraint, defined by a convex cone K. By transferring the local decomposition approach, we define strong stationarity conditions and provide a constraint qualification, under which these conditions are necessary for optimality. To apply this technique, we provide a new uniqueness result for Lagrange multipliers in Banach spaces. In the case that the cone K is polyhedral, we show that our strong stationarity conditions possess a reasonable strength. Finally, we generalize to the case where K is not a cone and apply the theory to two examples.}, language = {en} } @misc{HerzogObermeierWachsmuth, author = {Herzog, Roland and Obermeier, Johannes and Wachsmuth, Gerd}, title = {Annular and sectorial sparsity in optimal control of elliptic equations}, series = {Computational Optimization and Applications (COAP)}, volume = {62}, journal = {Computational Optimization and Applications (COAP)}, number = {1}, issn = {0926-6003}, doi = {10.1007/s10589-014-9721-5}, pages = {157 -- 180}, abstract = {Optimal control problems are considered with linear elliptic equations in polar coordinates. The objective contains L¹-type norms, which promote sparse optimal controls. The particular iterated structure of these norms gives rise to either annular or sectorial sparsity patterns. Optimality conditions and numerical solution approaches are developed.}, language = {en} } @misc{CasasHerzogWachsmuth, author = {Casas, Eduardo and Herzog, Roland and Wachsmuth, Gerd}, title = {Analysis of Spatio-Temporally Sparse Optimal Control Problems of Semilinear Parabolic Equations}, series = {ESAIM: Control, Optimisation and Calculus of Variations (COCV)}, volume = {23}, journal = {ESAIM: Control, Optimisation and Calculus of Variations (COCV)}, number = {1}, issn = {1292-8119}, doi = {10.1051/cocv/2015048}, pages = {263 -- 295}, abstract = {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.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Optimal Control of Quasistatic Plasticity with Linear Kinematic Hardening III: Optimality Conditions}, series = {Zeitschrift f{\"u}r Analysis und ihre Anwendungen}, volume = {35}, journal = {Zeitschrift f{\"u}r Analysis und ihre Anwendungen}, number = {1}, issn = {0232-2064}, doi = {10.4171/ZAA/1556}, pages = {81 -- 118}, abstract = {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.}, language = {en} } @misc{MehlitzWachsmuth, author = {Mehlitz, Patrick and Wachsmuth, Gerd}, title = {Weak and strong stationarity in generalized bilevel programming and bilevel optimal control}, series = {Optimization}, volume = {65}, journal = {Optimization}, number = {5}, issn = {0233-1934}, doi = {10.1080/02331934.2015.1122007}, pages = {907 -- 935}, abstract = {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.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Strong stationarity for optimization problems with complementarity constraints in absence of polyhedricity}, series = {Set-Valued and Variational Analysis}, volume = {25}, journal = {Set-Valued and Variational Analysis}, number = {1}, issn = {1877-0533}, doi = {10.1007/s11228-016-0370-y}, pages = {133 -- 175}, abstract = {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.}, language = {en} } @misc{SchneiderWachsmuth, author = {Schneider, Ren{\´e} and Wachsmuth, Gerd}, title = {A-posteriori error estimation for control-constrained, linear-quadratic optimal control problems}, series = {SIAM Journal on Numerical Analysis}, volume = {54}, journal = {SIAM Journal on Numerical Analysis}, number = {2}, issn = {1095-7170}, doi = {10.1137/15M1020460}, pages = {1169 -- 1192}, abstract = {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.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Towards M-stationarity for optimal control of the obstacle problem with control constraints}, series = {SIAM Journal on Control and Optimization}, volume = {54}, journal = {SIAM Journal on Control and Optimization}, number = {2}, issn = {1095-7138}, doi = {10.1137/140980582}, pages = {964 -- 986}, abstract = {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.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Pointwise Constraints in Vector-Valued Sobolev Spaces. With Applications in Optimal Control}, series = {Applied Mathematics \& Optimization}, volume = {77}, journal = {Applied Mathematics \& Optimization}, number = {3}, issn = {0095-4616}, doi = {10.1007/s00245-016-9381-1}, pages = {463 -- 497}, abstract = {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.}, language = {en} } @misc{MehlitzWachsmuth, author = {Mehlitz, Patrick and Wachsmuth, Gerd}, title = {The limiting normal cone to pointwise defined sets in Lebesgue spaces}, series = {Set-Valued and Variational Analysis (SVAA)}, volume = {26}, journal = {Set-Valued and Variational Analysis (SVAA)}, number = {3}, issn = {1877-0533}, doi = {10.1007/s11228-016-0393-4}, pages = {449 -- 467}, abstract = {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.}, language = {en} } @misc{RoeschWachsmuth, author = {R{\"o}sch, Arnd and Wachsmuth, Gerd}, title = {Mass lumping for the optimal control of elliptic partial differential equations}, series = {SIAM Journal on Numerical Analysis (SINUM)}, volume = {55}, journal = {SIAM Journal on Numerical Analysis (SINUM)}, number = {3}, issn = {0036-1429}, doi = {10.1137/16M1074473}, pages = {1412 -- 1436}, abstract = {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.}, language = {en} } @misc{GonzalezMerinoJahnPolyanskiietal., author = {Gonz{\´a}lez Merino, Bernardo and Jahn, Thomas and Polyanskii, Alexander and Wachsmuth, Gerd}, title = {Hunting for Reduced Polytopes}, series = {Discrete \& Computational Geometry}, volume = {60}, journal = {Discrete \& Computational Geometry}, number = {3}, issn = {0179-5376}, doi = {10.1007/s00454-018-9982-3}, pages = {801 -- 808}, abstract = {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.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {A guided tour of polyhedric sets: basic properties, new results on intersections and applications}, series = {Journal of Convex Analysis}, volume = {26}, journal = {Journal of Convex Analysis}, number = {1}, issn = {0944-6532}, pages = {153 -- 188}, abstract = {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.}, language = {en} } @misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {No-Gap Second-Order Conditions via a Directional Curvature Functional}, series = {SIAM Journal on Optimization}, volume = {28}, journal = {SIAM Journal on Optimization}, number = {3}, issn = {1052-6234}, doi = {10.1137/17M1140418}, pages = {2097 -- 2130}, abstract = {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.}, language = {en} }