@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{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{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{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{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{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{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{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{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{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{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{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{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{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{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{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{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{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{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{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{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{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 = {Non-existence of perturbed solutions under a second-order sufficient condition}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 4}, language = {en} } @techreport{AlphonseWachsmuth, author = {Alphonse, Amal and Wachsmuth, Gerd}, title = {Subdifferentials and penalty approximations of the obstacle problem}, series = {arXiv}, journal = {arXiv}, pages = {23}, 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 = {SIAM Journal on Control and Optimization}, volume = {62}, journal = {SIAM Journal on Control and Optimization}, number = {3}, doi = {10.1137/23m1586446}, pages = {1669 -- 1689}, language = {en} } @misc{HmedePaquetWachsmuth, author = {Hmede, Hiba and Paquet, Luc and Wachsmuth, Gerd}, title = {No-gap second-order optimality conditions for additive manufacturing}, series = {arXiv}, journal = {arXiv}, doi = {10.48550/arXiv.2410.18776}, pages = {27}, language = {en} } @misc{WachsmuthWalter, author = {Wachsmuth, Gerd and Walter, Daniel}, title = {No-gap second-order conditions for minimization problems in spaces of measures}, series = {arXiv}, journal = {arXiv}, doi = {10.48550/arXiv.2403.12001}, pages = {36}, 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 = {Optimization}, journal = {Optimization}, doi = {10.1080/02331934.2024.2328648}, pages = {19}, 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{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{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{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{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{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{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{KanzowKraemerMehlitzetal., author = {Kanzow, Christian and Kr{\"a}mer, Fabius and Mehlitz, Patrick and Wachsmuth, Gerd and Werner, Frank}, title = {Variational Poisson denoising via augmented Lagrangian methods}, series = {ETNA - Electronic Transactions on Numerical Analysis}, volume = {63}, journal = {ETNA - Electronic Transactions on Numerical Analysis}, doi = {10.1553/etna_vol63s33}, pages = {33 -- 62}, 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 = {SIAM Journal on Control and Optimization}, volume = {63}, journal = {SIAM Journal on Control and Optimization}, number = {1}, doi = {10.1137/23m1597332}, pages = {292 -- 320}, language = {en} } @misc{BorchardWachsmuth, author = {Borchard, Nicolas and Wachsmuth, Gerd}, title = {Second-order conditions for spatio-temporally sparse optimal control via second subderivatives}, series = {Journal of Nonsmooth Analysis and Optimizaton}, volume = {5}, journal = {Journal of Nonsmooth Analysis and Optimizaton}, doi = {10.46298/jnsao-2024-12604}, pages = {1 -- 36}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {A convex, finite and lower semicontinuous function with empty subdifferential}, series = {Journal of convex analysis}, volume = {32}, journal = {Journal of convex analysis}, number = {3}, publisher = {Heldermann Verlag}, address = {Lemgo}, pages = {877 -- 882}, abstract = {We give an example of a convex, finite and lower semicontinuous function whose subdifferential is everywhere empty. This is possible since the function is defined on an incomplete normed space. The function serves as a universal counterexample to various statements in convex analysis in which completeness is required.}, language = {en} } @misc{MarkoWachsmuth, author = {Marko, Jonas and Wachsmuth, Gerd}, title = {Vector-valued integer optimal control with TV regularization : optimality conditions and algorithmic treatment}, series = {ESAIM : control, optimisation and calculus of variations}, volume = {31}, journal = {ESAIM : control, optimisation and calculus of variations}, publisher = {EDP Sciences}, address = {Les Ulis}, issn = {1292-8119}, doi = {10.1051/cocv/2025037}, pages = {1 -- 23}, abstract = {We investigate a broad class of integer optimal control problems with vector-valued controls and switching regularization using a total variation functional involving the p-norm, which influences the structure of a solution. We derive optimality conditions of first and second order for the integer optimal control problem via a switching-point reformulation. For the numerical solution, we use a trust-region method utilizing Bellman's optimality principle for the subproblems. We will show convergence properties of the method and highlight the algorithm's efficacy on some benchmark examples.}, language = {en} } @misc{AlphonseWachsmuth, author = {Alphonse, Amal and Wachsmuth, Gerd}, title = {Subdifferentials and penalty approximations of the obstacle problem}, series = {SIAM journal on optimization}, volume = {35}, journal = {SIAM journal on optimization}, number = {3}, publisher = {SIAM}, address = {Philadelphia, Pa.}, doi = {10.1137/24m172202x}, pages = {2017 -- 2039}, abstract = {We consider a framework for approximating the obstacle problem through a penalty approach by nonlinear PDEs. By using tools from capacity theory, we show that derivatives of the solution maps of the penalized problems converge in the weak operator topology to an element of the strong-weak Bouligand subdifferential. We are able to treat smooth penalty terms as well as nonsmooth ones involving, for example, the positive part function max⁡(0, ⋅). Our abstract framework applies to several specific choices of penalty functions which are omnipresent in the literature. We conclude with consequences to the theory of optimal control of the obstacle problem.}, language = {en} } @misc{FriedemannWachsmuth, author = {Friedemann, Markus and Wachsmuth, Gerd}, title = {A trust-region method for optimal control of ODEs with continuous-or-off controls and TV regularization}, series = {arXiv}, journal = {arXiv}, publisher = {arXiv}, doi = {10.48550/arXiv.2508.10692}, pages = {1 -- 50}, abstract = {A solution algorithm for a special class of optimal control problems subject to an ordinary differential equation is proposed. The controls possess a continuous-or-off structure and are priced by a convex function. Additionally a total variation regularization is applied to penalize switches. Our solution method combines a trust-region method and a proximal gradient method. The subproblems are solved via Bellman's optimality principle. Convergence with respect to a criticality measure is proven. As a numerical example, we solve a simple optimal control problem involving an SIR model.}, language = {en} } @misc{MeyerWachsmuth, author = {Meyer, Christian and Wachsmuth, Gerd}, title = {Optimal control of the Poisson equation with transport regularization : properties of optimal transport plans and transport map}, series = {arXiv}, journal = {arXiv}, publisher = {arXiv}, doi = {10.48550/arXiv.2506.02808}, pages = {1 -- 37}, abstract = {An optimal control problem in the space of Borel measures governed by the Poisson equation is investigated. The characteristic feature of the problem under consideration is the Tikhonov regularization term in form of the transportation distance of the control to a given prior. Existence of optimal solutions is shown and first-order necessary optimality conditions are derived. The latter are used to deduce structural a priori information about the optimal control and its support based on properties of the associated optimal transport plan.}, language = {en} } @misc{BorchardWachsmuth, author = {Borchard, Nicolas and Wachsmuth, Gerd}, title = {Numerical solution of optimal control problems using quadratic transport regularization}, series = {arXiv}, journal = {arXiv}, publisher = {arXiv}, doi = {10.48550/arXiv.2503.07105}, pages = {1 -- 29}, abstract = {We address optimal control problems on the space of measures for an objective containing a smooth functional and an optimal transport regularization. That is, the quadratic Monge-Kantorovich distance between a given prior measure and the control is penalized in the objective. We consider optimality conditions and reparametrize the problem using the celebrated structure theorem by Brenier. The optimality conditions can be formulated as a piecewise differentiable equation. This is utilized to formulate solution algorithms and to analyze their local convergence properties. We present a numerical example to illustrate the theoretical findings.}, language = {en} } @misc{BorchardWachsmuth, author = {Borchard, Nicolas and Wachsmuth, Gerd}, title = {Characterization of Hilbertizable spaces via convex functions}, series = {arXiv}, journal = {arXiv}, publisher = {arXiv}, doi = {10.48550/arXiv.2506.04686}, pages = {1 -- 8}, abstract = {We show that the existence of a strongly convex function with a Lipschitz derivative on a Banach space already implies that the space is isomorphic to a Hilbert space. Similarly, if both a function and its convex conjugate are then the underlying space is also isomorphic to a Hilbert space.}, language = {en} } @misc{WachsmuthWalter, author = {Wachsmuth, Gerd and Walter, Daniel}, title = {Proximal gradient methods in Banach spaces}, series = {arXiv}, journal = {arXiv}, publisher = {arXiv}, doi = {10.48550/arXiv.2509.24685}, pages = {1 -- 39}, abstract = {Proximal gradient methods are a popular tool for the solution of structured, nonsmooth minimization problems. In this work, we investigate an extension of the former to general Banach spaces and provide worst-case convergence rates for, both, convex and nonconvex, problem instances. Moreover, assuming additional regularity properties of stationary points, linear rates of convergence are derived. The theoretical results are illustrated for bang-bang type optimal control problems with partial differential equations which we study in the space of Radon measures. An efficient implementation of the resulting -proximal gradient method is given and its performance is compared to standard -proximal gradient as well as Frank-Wolfe methods. The paper is complemented by discussing the relationship among different regularity properties as well as by providing a novel characterization of the Polyak--Łojasiewicz--Kurdyka property via second-order conditions involving weak* second subderivatives.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {New numerical solutions to Newton's problem of least resistance via a convex hull approach}, series = {arXiv}, journal = {arXiv}, publisher = {arXiv}, address = {Ithaca, NY}, doi = {https://doi.org/10.48550/arXiv.2511.09177}, pages = {1 -- 12}, abstract = {We present a numerical method for the solution of Newton's problem of least resistance in the class of convex functions using a convex hull approach. We observe that the numerically computed solutions possess some symmetry. Further, their extremal points lie on several curves. By exploiting this conjectured structure, we are able to compute highly accurate solutions to Newton's problem.}, language = {en} } @misc{HmedePaquetWachsmuth, author = {Hmede, Hiba and Paquet, Luc and Wachsmuth, Gerd}, title = {No-gap second-order optimality conditions for additive manufacturing}, series = {Mathematical control and related fields}, journal = {Mathematical control and related fields}, publisher = {AIMS}, address = {Springfield, MO}, issn = {2156-8499}, doi = {10.3934/mcrf.2025028}, pages = {1 -- 29}, abstract = {Additive manufacturing by laser fusion on a metal oxides powder bed has developed considerably in the last few years and allows to produce a wide range of complex parts. The mathematical models correspond to initial boundary value problems for the heat equation with moving heat sources according to the laser trajectories. The main questions concern the optimization of the trajectories scanned by the laser and of the thermal treatment time in order to melt the powder where it is desired to make the part and to minimize the thermal gradients. Our purpose in this current paper is to pursue the study of the optimization model that we have introduced in a previous paper. Here, we consider second-order optimality conditions for non-necessarily convex constraints on the laser paths. In particular, we obtain no gap between the second-order sufficient optimality condition and the necessary second-order optimality condition. To achieve this goal, we reformulate our optimal control problem in order to fit it in the framework of the abstract theory of optimization under constraints in Banach spaces. Higher regularity of the trajectories for local minimizers is also proved implying higher regularity of the corresponding Lagrange multipliers. The case of the regularity of the trajectories for stationary points is left open.}, 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 = {Interfaces and free boundaries : mathematical analysis, computation and applications}, volume = {27}, journal = {Interfaces and free boundaries : mathematical analysis, computation and applications}, number = {4}, publisher = {EMS Press}, address = {Z{\"u}rich}, issn = {1463-9971}, doi = {10.4171/ifb/545}, pages = {521 -- 573}, abstract = {Quasi-variational inequalities (QVIs) of obstacle type in many cases have multiple solutions that can be ordered. We study a multitude of properties of the operator mapping the source term to the minimal or maximal solution of such QVIs. We prove that the solution maps are locally Lipschitz continuous and directionally differentiable and show existence of optimal controls for problems that incorporate these maps as the control-to-state operator. We also consider a Moreau-Yosida-type penalisation for the QVI, wherein we show that it is possible to approximate the minimal and maximal solutions by sequences of minimal and maximal solutions (respectively) of certain PDEs, which have a simpler structure and offer a convenient characterisation in particular for computation. For solution mappings of these penalised problems, we prove a number of properties including Lipschitz and differential stability. Making use of the penalised equations, we derive (in the limit) C-stationarity conditions for the control problem, in addition to the Bouligand stationarity we get from the differentiability result.}, language = {en} }