@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{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{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{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{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{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{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} }