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Gerd Wachsmuth

Publications and source records attributed to Gerd Wachsmuth.

At least 19 recordsLinked to original sources
arXiv2026

No-gap second-order conditions for optimization problems involving transport distances

We consider optimization problems in the space of measures. As a regularization term, the problem includes the transport distance to a given prior measure. For the derivation of second-order optimality conditions of no-gap type, the theory of weak-$\star$ second subderivatives is used which will lead to an equivalence with quadratic growth under additional assumptions on the smooth part of the objective and on the Kantorovich potential, i.e., the solution of the dual transport problem. Further, the weak-$\star$ second subderivative is calculated and weak-$\star$ epidifferentiability is proven. Finally, the results are applied to optimal control problems in measure space.

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arXiv2026

Uniqueness and stability of Lagrange multipliers and associated qualification conditions

This paper is concerned with uniqueness and stability of Lagrange multipliers for constrained optimization problems in abstract spaces. It is well known that validity of the strict Robinson-Zowe-Kurcyusz condition implies the so-called isolated calmness, a one-sided Lipschitz property tailored for set-valued mappings, of some Lagrange multiplier mapping associated with a perturbed version of the original optimization problem, and the latter indeed is enough to guarantee uniqueness of the Lagrange multiplier. The paper studies the isolated calmness of the Lagrange multiplier mapping in detail. Exemplary, it is shown that this condition is sufficient for the Robinson-Zowe-Kurcyusz constraint qualification and, in the presence of additional assumptions, even equivalent to the strict Robinson-Zowe-Kurcyusz condition. Illustrative examples are presented to underline the necessity of postulated assumptions.

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arXiv2026

Second-order conditions for bang-bang control of elliptic equations in arbitrary dimensions

We consider an optimal control problem governed by a semilinear PDE in cases where the optimal control is of bang-bang type. By utilizing the theory of Bessel potential space, we characterize quadratic growth of the objective via a second-order optimality condition. In contrast to previous contributions, our method of proof works in arbitrary spatial dimensions.

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arXiv2025

New numerical solutions to Newton's problem of least resistance via a convex hull approach

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.

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arXiv2025

Proximal gradient methods in Banach spaces

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 $L^1$-proximal gradient method is given and its performance is compared to standard $L^2$-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--{\L}ojasiewicz--Kurdyka property via second-order conditions involving weak* second subderivatives.

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arXiv2025

A trust-region method for optimal control of ODEs with continuous-or-off controls and TV regularization

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.

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arXiv2025

Characterization of Hilbertizable spaces via convex functions

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 $C^2$ then the underlying space is also isomorphic to a Hilbert space.

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arXiv2025

Optimal control of the Poisson equation with transport regularization: Properties of optimal transport plans and transport map

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.

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arXiv2025

Numerical solution of optimal control problems using quadratic transport regularization

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.

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arXiv2024

Subdifferentials and penalty approximations of the obstacle problem

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 penalised 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 $\max(0,\cdot)$. 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.

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arXiv2024

Vector-Valued Integer Optimal Control with TV Regularization: Optimality Conditions and Algorithmic Treatment

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 algorithms efficacy on some benchmark examples.

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arXiv2024

No-gap second-order optimality conditions for additive manufacturing

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.

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arXiv2024

A convex, finite and lower semicontinuous function with empty subdifferential

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.

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arXiv2024

Variational Poisson Denoising via Augmented Lagrangian Methods

In this paper, we denoise a given noisy image by minimizing a smoothness promoting function over a set of local similarity measures which compare the mean of the given image and some candidate image on a large collection of subboxes. The associated convex optimization problem possesses a huge number of constraints which are induced by extended real-valued functions stemming from the Kullback--Leibler divergence. Alternatively, these nonlinear constraints can be reformulated as affine ones, which makes the model seemingly more tractable. For the numerical treatment of both formulations of the model (i.e., the original one as well as the one with affine constraints), we propose a rather general augmented Lagrangian method which is capable of handling the huge amount of constraints. A self-contained, derivative-free, global convergence theory is provided, allowing an extension to other problem classes. For the solution of the resulting subproblems in the setting of our suggested image denoising models, we make use of a suitable stochastic gradient method. Results of several numerical experiments are presented in order to compare both formulations and the associated augmented Lagrangian methods.

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arXiv2024

No-gap second-order conditions for minimization problems in spaces of measures

Over the last years, minimization problems over spaces of measures have received increased interest due to their relevance in the context of inverse problems, optimal control and machine learning. A fundamental role in their numerical analysis is played by the assumption that the optimal dual state admits finitely many global extrema and satisfies a second-order sufficient optimality condition in each one of them. In this work, we show the full equivalence of these structural assumptions to a no-gap second-order condition involving the second subderivative of the Radon norm as well as to a local quadratic growth property of the objective functional with respect to the bounded Lipschitz norm.

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arXiv2024

Slater conditions without interior points for programs in Lebesgue spaces with pointwise bounds and finitely many constraints

We consider optimization problems in Lebesgue spaces with pointwise box constraints and finitely many additional linear constraints. We prove that the existence of a Slater point which lies strictly between the pointwise bounds and which satisfies the linear constraints is sufficient for the existence of Lagrange multipliers. Surprisingly, the Slater point is also necessary for the existence of Lagrange multipliers in a certain sense. We also demonstrate how to handle additional finitely many nonlinear constraints.

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arXiv2023

Minimal and maximal solution maps of elliptic QVIs of obstacle type: Lipschitz stability, differentiability and optimal control

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.

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