@misc{BoergensKanzowMehlitzetal., author = {B{\"o}rgens, Eike and Kanzow, Christian and Mehlitz, Patrick and Wachsmuth, Gerd}, title = {New Constraint Qualifications for Optimization Problems in Banach Spaces Based on Asymptotic KKT Conditions}, series = {SIAM Journal on Optimization}, volume = {30}, journal = {SIAM Journal on Optimization}, number = {4}, issn = {1095-7189}, doi = {10.1137/19M1306804}, pages = {2956 -- 2982}, abstract = {Optimization theory in Banach spaces suffers from a lack of available constraint qualifications. There exist very few constraint qualifications, and these are often violated even in simple applications. This is very much in contrast to finite-dimensional nonlinear programs, where a large number of constraint qualifications is known. Since these constraint qualifications are usually defined using the set of active inequality constraints, it is difficult to extend them to the infinite-dimensional setting. One exception is a recently introduced sequential constraint qualification based on asymptotic KKT conditions. This paper shows that this so-called asymptotic KKT regularity allows suitable extensions to the Banach space setting in order to obtain new constraint qualifications. The relation of these new constraint qualifications to existing ones is discussed in detail. Their usefulness is also shown by several examples as well as an algorithmic application to the class of augmented Lagrangian methods.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {No-Gap Second-Order Conditions under n-Polyhedric Constraints and Finitely Many Nonlinear Constraints}, series = {Journal of Convex Analysis}, volume = {27}, journal = {Journal of Convex Analysis}, number = {2}, issn = {2363-6394}, pages = {733 -- 751}, abstract = {We consider an optimization problem subject to an abstract constraint and finitely many nonlinear constraints. Using the recently introduced concept of n-polyhedricity, we are able to provide second-order optimality conditions under weak regularity assumptions. In particular, we prove necessary optimality conditions of first and second order under the constraint qualification of Robinson, Zowe and Kurcyusz. Similarly, sufficient optimality conditions are stated. The gap between both conditions is as small as possible.}, language = {en} } @misc{ChristofWachsmuth, author = {Christof, Constantin and Wachsmuth, Gerd}, title = {On Second-Order Optimality Conditionsfor Optimal Control Problems Governedby the Obstacle Problem}, series = {Optimization}, journal = {Optimization}, issn = {1029-4945}, doi = {10.1080/02331934.2020.1778686}, pages = {41}, abstract = {This paper is concerned with second-order optimality conditions for Tikhonov regularized optimal control problems governed by the obstacle problem. Using a simple observation that allows to characterize the structure of optimal controls on the active set, we derive various conditions that guarantee the local/global optimality of first-order stationary points and/or the local/global quadratic growth of the reduced objective function. Our analysis extends and refines existing results from the literature and also covers those situations where the problem at hand involves additional box-constraints on the control. As a byproduct, our approach shows in particular that Tikhonov regularized optimal control problems for the obstacle problem can be reformulated as state-constrained optimal control problems for the Poisson equation and that problems involving a subharmonic obstacle and a convex objective function are uniquely solvable. The paper concludes with three counterexamples which illustrate that rather peculiar effects can occur in the analysis of second-order optimality conditions for optimal control problems governed by the obstacle problem and that necessary second-order conditions for such problems may be hard to derive.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Elliptic quasi-variational inequalities under a smallness assumption: uniqueness, differential stability and optimal control}, series = {Calculus of Variations and Partial Differential Equations}, volume = {59}, journal = {Calculus of Variations and Partial Differential Equations}, number = {2}, issn = {1432-0835}, doi = {10.1007/s00526-020-01743-3}, pages = {15}, abstract = {We consider a quasi-variational inequality governed by a moving set. We employ the assumption that the movement of the set has a small Lipschitz constant. Under this requirement, we show that the quasi-variational inequality has a unique solution which depends Lipschitz-continuously on the source term. If the data of the problem is (directionally) differentiable, the solution map is directionally differentiable as well. We also study the optimal control of the quasi-variational inequality and provide necessary optimality conditions of strongly stationary type.}, language = {en} } @misc{EtlingHerzogLoayzaetal., author = {Etling, Tommy and Herzog, Roland and Loayza, Estefania and Wachsmuth, Gerd}, title = {First and Second Order Shape Optimization Based on Restricted Mesh Deformations}, series = {SIAM Journal on Scientific Computing}, volume = {42}, journal = {SIAM Journal on Scientific Computing}, number = {2}, issn = {1095-7197}, doi = {10.1137/19M1241465}, pages = {A1200 -- A1225}, abstract = {We consider shape optimization problems subject to elliptic partial differential equations. In the context of the finite element method, the geometry to be optimized is represented by the computational mesh, and the optimization proceeds by repeatedly updating the mesh node positions. It is well known that such a procedure eventually may lead to a deterioration of mesh quality, or even an invalidation of the mesh, when interior nodes penetrate neighboring cells. We examine this phenomenon, which can be traced back to the ineptness of the discretized objective when considered over the space of mesh node positions. As a remedy, we propose a restriction in the admissible mesh deformations, inspired by the Hadamard structure theorem. First and second order methods are considered in this setting. Numerical results show that mesh degeneracy can be overcome, avoiding the need for remeshing or other strategies. FEniCS code for the proposed methods is available on GitHub.}, language = {en} } @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} } @incollection{MehlitzWachsmuth, author = {Mehlitz, Patrick and Wachsmuth, Gerd}, title = {Bilevel optimal control: existence results and stationarity conditions}, series = {Bilevel Optimization}, booktitle = {Bilevel Optimization}, publisher = {Springer Nature}, address = {Schweiz}, isbn = {978-3-030-52119-6}, doi = {10.1007/978-3-030-52119-6_16}, pages = {451 -- 484}, abstract = {The mathematical modeling of numerous real-world applications results in hierarchical optimization problems with two decision makers where at least one of them has to solve an optimal control problem of ordinary or partial differential equations. Such models are referred to as bilevel optimal control problems. Here, we first review some different features of bilevel optimal control including important applications, existence results, solution approaches, and optimality conditions. Afterwards, we focus on a specific problem class where parameters appearing in the objective functional of an optimal control problem of partial differential equations have to be reconstructed. After verifying the existence of solutions, necessary optimality conditions are derived by exploiting the optimal value function of the underlying parametric optimal control problem in the context of a relaxation approach.}, 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} } @incollection{HerzogMeyerWachsmuth, author = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd}, title = {Optimal Control of Elastoplastic Processes: Analysis, Algorithms, Numerical Analysis and Applications}, series = {Trends in PDE Constrained Optimization}, booktitle = {Trends in PDE Constrained Optimization}, editor = {Leugering, G{\"u}nter}, edition = {1. Auflage}, publisher = {Birkh{\"a}user}, address = {Cham}, isbn = {978-3-319-05082-9}, doi = {10.1007/978-3-319-05083-6_4}, pages = {27 -- 41}, abstract = {An optimal control problem is considered for the variational inequality representing the stress-based (dual) formulation of static elastoplasticity. The linear kinematic hardening model and the von Mises yield condition are used. The forward system is reformulated such that it involves the plastic multiplier and a complementarity condition. In order to derive necessary optimality conditions, a family of regularized optimal control problems is analyzed. C-stationarity type conditions are obtained by passing to the limit with the regularization. Numerical results are presented.}, language = {en} } @misc{HarderWachsmuth, author = {Harder, Felix and Wachsmuth, Gerd}, title = {M-stationarity for a class of MPCCs in Lebesgue spaces}, series = {arXiv}, journal = {arXiv}, abstract = {We show that an optimality condition of M-stationarity type holds for minimizers of a class of mathematical programs with complementarity constraints (MPCCs) in Lebesgue spaces. We apply these results also to local minimizers of an inverse optimal control problem (which is an instance of an infinite-dimensional bilevel optimization problem). The multipliers for the M-stationarity system can be constructed via convex combinations of various multipliers to auxiliary, linear problems. However, proving the existence of the multipliers to these auxiliary problems is difficult and only possible in some situations.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 27}, abstract = {We consider a generalized equation governed by a strongly monotone and Lipschitz single-valued mapping and a maximally monotone set-valued mapping in a Hilbert space. We are interested in the sensitivity of solutions w.r.t. perturbations of both mappings. We demonstrate that the directional differentiability of the solution map can be verified by using the directional differentiability of the single-valued operator and of the resolvent of the set-valued mapping. The result is applied to quasi-generalized equations in which we have an additional dependence of the solution within the set-valued part of the equation.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Maximal monotone operators with non-maximal graphical limit}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 4}, abstract = {We present a counterexample showing that the graphical limit of maximally monotone operators might not be maximally monotone. We also characterize the directional differentiability of the resolvent of an operator B in terms of existence and maximal monotonicity of the proto-derivative of B.}, language = {en} } @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{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} }