TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces T2 - Journal of Convex Analysis Y1 - 2023 UR - https://www.heldermann.de/JCA/JCA30/JCA304/jca30059.htm VL - 30 IS - 4 SP - 1319 EP - 1328 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Continuous differentiability of the signum function and Newton's method for bang-bang control T2 - arXiv N2 - We investigate bang-bang control problems and the possibility to apply Newton's method to solve such kind of problems numerically. To this end, we show that the signum function is Fréchet differentiable between appropriate function spaces. Numerical experiments show the applicability of the resulting method. KW - Bang-bang controls KW - Newton method KW - Signum function Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2509.24829 SP - 1 EP - 28 PB - arXiv CY - Ithaca, NY 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 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 -