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 -