TY - GEN A1 - Domínguez Corella, Alberto A1 - Wachsmuth, Gerd T1 - Stability and genericity of bang-bang controls in affine problems T2 - SIAM Journal on Control and Optimization Y1 - 2024 U6 - https://doi.org/10.1137/23m1586446 VL - 62 IS - 3 SP - 1669 EP - 1689 ER - TY - GEN A1 - Hmede, Hiba A1 - Paquet, Luc A1 - Wachsmuth, Gerd T1 - No-gap second-order optimality conditions for additive manufacturing T2 - arXiv Y1 - 2024 UR - https://arxiv.org/abs/2410.18776 U6 - https://doi.org/10.48550/arXiv.2410.18776 ER - TY - GEN A1 - Wachsmuth, Gerd A1 - Walter, Daniel T1 - No-gap second-order conditions for minimization problems in spaces of measures T2 - arXiv Y1 - 2024 UR - https://arxiv.org/abs/2403.12001 U6 - https://doi.org/10.48550/arXiv.2403.12001 ER - TY - GEN A1 - Wachsmuth, Gerd T1 - Slater conditions without interior points for programs in Lebesgue spaces with pointwise bounds and finitely many constraints T2 - Optimization Y1 - 2024 U6 - https://doi.org/10.1080/02331934.2024.2328648 ER - TY - GEN A1 - Alphonse, Amal A1 - Hintermüller, Michael A1 - Rautenberg, Carlos N. A1 - Wachsmuth, Gerd T1 - Minimal and maximal solution maps of elliptic QVIs: penalisation, Lipschitz stability, differentiability and optimal control T2 - arXiv Y1 - 2023 UR - https://arxiv.org/abs/2312.13879 SP - 1 EP - 33 ER - TY - GEN A1 - Borchard, Nicolas A1 - Wachsmuth, Gerd T1 - Second-order conditions for spatio-temporally sparse optimal control via second subderivatives T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2311.14538 SP - 1 EP - 42 ER - TY - GEN A1 - Bartels, Sören A1 - Keller, Hedwig A1 - Wachsmuth, Gerd T1 - Numerical Approximation of Optimal Convex Shapes in ℝ³ T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2311.13386 SP - 1 EP - 17 ER - TY - GEN A1 - Christof, Constantin A1 - Wachsmuth, Gerd T1 - Energy Space Newton Differentiability for Solution Maps of Unilateral and Bilateral Obstacle Problems T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2308.15289 SP - 1 EP - 26 ER - TY - GEN A1 - Domínguez Corella, Alberto A1 - Wachsmuth, Gerd T1 - Stability and genericity of bang-bang controls in affine problems T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2307.05418 SP - 1 EP - 25 ER - TY - GEN A1 - Dempe, Stephan A1 - Friedemann, Markus A1 - Harder, Felix A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - Bilevel optimal control: theory, algorithms, and applications T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2305.19786 SP - 1 EP - 31 ER - TY - GEN A1 - Kanzow, Christian A1 - Krämer, Fabius A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd A1 - Werner, Frank T1 - A nonsmooth augmented Lagrangian method and its application to Poisson denoising and sparse control T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2304.06434 SP - 1 EP - 36 ER - TY - GEN A1 - Keller, Hedwig A1 - Bartels, Sören A1 - Wachsmuth, Gerd T1 - Numerical Approximation of Optimal Convex and Rotationally Symmetric Shapes for an Eigenvalue Problem arising in Optimal Insulation T2 - Computers & Mathematics with Applications Y1 - 2022 U6 - https://doi.org/10.1016/j.camwa.2022.05.026 SN - 1873-7668 SN - 0898-1221 VL - 119 SP - 327 EP - 339 ER - TY - GEN A1 - Wachsmuth, Gerd T1 - Non-existence of perturbed solutions under a second-order sufficient condition T2 - Examples and Counterexamples Y1 - 2023 U6 - https://doi.org/10.1016/j.exco.2023.100122 SN - 2666-657X VL - 4 SP - 1 EP - 3 ER - TY - GEN A1 - Marko, Jonas A1 - Wachsmuth, Gerd T1 - Integer optimal control problems with total variation regularization: Optimality conditions and fast solution of subproblems T2 - ESAIM: Control, Optimisation and Calculus of Variations Y1 - 2023 U6 - https://doi.org/10.1051/cocv/2023065 VL - 29 SP - 1 EP - 30 ER - TY - GEN A1 - Christof, Constantin A1 - Wachsmuth, Gerd T1 - Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control T2 - SIAM Journal on Control and Optimization Y1 - 2023 U6 - https://doi.org/10.1137/21m1467365 SN - 1095-7138 SN - 0887-4603 VL - 61 IS - 3 SP - 1162 EP - 1186 ER - TY - GEN A1 - Friedemann, Markus A1 - Harder, Felix A1 - Wachsmuth, Gerd T1 - Finding global solutions of some inverse optimal control problems using penalization and semismooth Newton methods T2 - Journal of Global Optimization Y1 - 2023 U6 - https://doi.org/10.1007/s10898-023-01288-7 SN - 1573-2916 SN - 0925-5001 VL - 86 IS - 4 SP - 1025 EP - 1061 ER - TY - GEN A1 - Kanzow, Christian A1 - Krämer, Fabius A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd A1 - Werner, Frank T1 - Variational Poisson denoising via augmented Lagrangian methods T2 - ETNA - Electronic Transactions on Numerical Analysis Y1 - 2025 U6 - https://doi.org/10.1553/etna_vol63s33 VL - 63 SP - 33 EP - 62 ER - TY - GEN A1 - Christof, Constantin A1 - Wachsmuth, Gerd T1 - Energy Space Newton Differentiability for Solution Maps of Unilateral and Bilateral Obstacle Problems T2 - SIAM Journal on Control and Optimization Y1 - 2025 U6 - https://doi.org/10.1137/23m1597332 VL - 63 IS - 1 SP - 292 EP - 320 ER - TY - GEN A1 - Borchard, Nicolas A1 - Wachsmuth, Gerd T1 - Second-order conditions for spatio-temporally sparse optimal control via second subderivatives T2 - Journal of Nonsmooth Analysis and Optimizaton Y1 - 2024 U6 - https://doi.org/10.46298/jnsao-2024-12604 VL - 5 SP - 1 EP - 36 ER - TY - GEN A1 - Wachsmuth, Gerd T1 - A convex, finite and lower semicontinuous function with empty subdifferential T2 - Journal of convex analysis N2 - 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. Y1 - 2025 UR - https://www.heldermann.de/JCA/JCA32/JCA323/jca32044.htm VL - 32 IS - 3 SP - 877 EP - 882 PB - Heldermann Verlag CY - Lemgo ER - TY - GEN A1 - Marko, Jonas A1 - Wachsmuth, Gerd T1 - Vector-valued integer optimal control with TV regularization : optimality conditions and algorithmic treatment T2 - ESAIM : control, optimisation and calculus of variations N2 - 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. KW - Integer optimal control problem KW - Total variation regularization KW - Trust-region method Y1 - 2025 U6 - https://doi.org/10.1051/cocv/2025037 SN - 1292-8119 SN - 1262-3377 VL - 31 SP - 1 EP - 23 PB - EDP Sciences CY - Les Ulis ER - TY - GEN A1 - Alphonse, Amal A1 - Wachsmuth, Gerd T1 - Subdifferentials and penalty approximations of the obstacle problem T2 - SIAM journal on optimization N2 - 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. KW - Variational inequality KW - Obstacle problem KW - Optimal control KW - Weak operator topology KW - Measures Y1 - 2025 U6 - https://doi.org/10.1137/24m172202x VL - 35 IS - 3 SP - 2017 EP - 2039 PB - SIAM CY - Philadelphia, Pa. ER - TY - GEN A1 - Friedemann, Markus A1 - Wachsmuth, Gerd T1 - A trust-region method for optimal control of ODEs with continuous-or-off controls and TV regularization T2 - arXiv N2 - 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. Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2508.10692 SP - 1 EP - 50 PB - arXiv ER - TY - GEN A1 - Meyer, Christian A1 - Wachsmuth, Gerd T1 - Optimal control of the Poisson equation with transport regularization : properties of optimal transport plans and transport map T2 - arXiv N2 - 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. KW - Optimal control of PDEs KW - Measure control KW - Optimal transport regularization KW - First-order necessary optimality conditions Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2506.02808 SP - 1 EP - 37 PB - arXiv ER - TY - GEN A1 - Borchard, Nicolas A1 - Wachsmuth, Gerd T1 - Numerical solution of optimal control problems using quadratic transport regularization T2 - arXiv N2 - 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. Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2503.07105 SP - 1 EP - 29 PB - arXiv ER - TY - GEN A1 - Borchard, Nicolas A1 - Wachsmuth, Gerd T1 - Characterization of Hilbertizable spaces via convex functions T2 - arXiv N2 - 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. KW - Hilbertizable space KW - strong convexity KW - Lipschitz continuous derivative KW - Convex KW - conjugate function Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2506.04686 SP - 1 EP - 8 PB - arXiv ER - TY - GEN A1 - Wachsmuth, Gerd A1 - Walter, Daniel T1 - Proximal gradient methods in Banach spaces T2 - arXiv N2 - 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. Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2509.24685 SP - 1 EP - 39 PB - arXiv ER - TY - GEN A1 - Wachsmuth, Gerd T1 - New numerical solutions to Newton's problem of least resistance via a convex hull approach T2 - arXiv N2 - 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. KW - Newton’s problem of minimal resistance KW - Convex hull KW - Calculus of variations with convexity constraint Y1 - 2025 UR - https://arxiv.org/abs/2511.09177 U6 - https://doi.org/https://doi.org/10.48550/arXiv.2511.09177 SP - 1 EP - 12 PB - arXiv CY - Ithaca, NY ER - TY - GEN A1 - Hmede, Hiba A1 - Paquet, Luc A1 - Wachsmuth, Gerd T1 - No-gap second-order optimality conditions for additive manufacturing T2 - Mathematical control and related fields N2 - 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. KW - Optimal control problems KW - Parabolic equations KW - Heat equation with moving heat sources KW - Time of thermal treatment KW - Cost functionals KW - Abstract theory of optimization under constraints in Banach spaces KW - Higher regularity of the trajectories for local minimizers KW - Lagrange multipliers KW - Constraint qualifications KW - Second-order optimality conditions KW - Quadratic Legendre forms KW - Polyhedric sets. Y1 - 2025 U6 - https://doi.org/10.3934/mcrf.2025028 SN - 2156-8499 SP - 1 EP - 29 PB - AIMS CY - Springfield, MO ER - TY - GEN A1 - Alphonse, Amal A1 - Hintermüller, Michael A1 - Rautenberg, Carlos N. A1 - Wachsmuth, Gerd T1 - Minimal and maximal solution maps of elliptic QVIs : penalisation, Lipschitz stability, differentiability and optimal control T2 - Interfaces and free boundaries : mathematical analysis, computation and applications N2 - 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. Y1 - 2025 U6 - https://doi.org/10.4171/ifb/545 SN - 1463-9971 VL - 27 IS - 4 SP - 521 EP - 573 PB - EMS Press CY - Zürich ER -