Searcharxiv⌕ Search

arXiv subjects

Gerd Wachsmuth

Publications and source records attributed to Gerd Wachsmuth.

At least 37 records · Page 2Linked to original sources
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.

math.OC↗
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.

math.OC↗
arXiv2023

Bilevel Optimal Control: Theory, Algorithms, and Applications

In this chapter, we are concerned with inverse optimal control problems, i.e., optimization models which are used to identify parameters in optimal control problems from given measurements. Here, we focus on linear-quadratic optimal control problems with control constraints where the reference control plays the role of the parameter and has to be reconstructed. First, it is shown that pointwise M-stationarity, associated with the reformulation of the hierarchical model as a so-called mathematical problem with complementarity constraints (MPCC) in function spaces, provides a necessary optimality condition under some additional assumptions on the data. Second, we review two recently developed algorithms (an augmented Lagrangian method and a nonsmooth Newton method) for the computational identification of M-stationary points of finite-dimensional MPCCs. Finally, a numerical comparison of these methods, based on instances of the appropriately discretized inverse optimal control problem of our interest, is provided.

math.OC↗
arXiv2023

Numerical Approximation of Optimal Convex Shapes in $\mathbb{R}^3$

In the optimization of convex domains under a PDE constraint numerical difficulties arise in the approximation of convex domains in $\mathbb{R}^3$. Previous research used a restriction to rotationally symmetric domains to reduce shape optimization problems to a two-dimensional setting. In the current research, two approaches for the approximation in $\mathbb{R}^3$ are considered. First, a notion of discrete convexity allows for a nearly convex approximation with polyhedral domains. An alternative approach is based on the recent observation that higher order finite elements can approximate convex functions conformally. As a second approach these results are used to approximate optimal convex domains with isoparametric convex domains. The proposed algorithms were tested on shape optimization problems constrained by a Poisson equation and both algorithms achieved similar results.

math.NA↗
arXiv2023

Energy Space Newton Differentiability for Solution Maps of Unilateral and Bilateral Obstacle Problems

We prove that the solution operator of the classical unilateral obstacle problem on a nonempty open bounded set $Ω\subset \mathbb{R}^d$, $d \in \mathbb{N}$, is Newton differentiable as a function from $L^p(Ω)$ to $H_0^1(Ω)$ whenever $\max(1, 2d/(d+2)) < p \leq \infty$. By exploiting this Newton differentiability property, results on angled subspaces in $H^{-1}(Ω)$, and a formula for orthogonal projections onto direct sums, we further show that the solution map of the classical bilateral obstacle problem is Newton differentiable as a function from $L^p(Ω)$ to $H_0^1(Ω)\cap L^q(Ω)$ whenever $\max(1, d/2) < p \leq \infty$ and $1 \leq q <\infty$. For both the unilateral and the bilateral case, we provide explicit formulas for the Newton derivative. As a concrete application example for our results, we consider the numerical solution of an optimal control problem with $H_0^1(Ω)$-controls and box-constraints by means of a semismooth Newton method.

math.OC↗
arXiv2023

Stability and genericity of bang-bang controls in affine problems

We analyse the role of the bang-bang property in affine optimal control problems. We show that many essential stability properties of affine problems are only satisfied when minimizers are bang-bang. Moreover, we prove that almost any perturbation in an affine optimal control problem leads to a bang-bang strict global minimizer. We work in an abstract framework that allows to cover many problems in the literature of optimal control, this includes problems constrained by partial and ordinary differential equations. We give examples that show the applicability of our results to specific optimal control problems.

math.OC↗
arXiv2023

Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control

We prove that solution operators of elliptic obstacle-type variational inequalities (or, more generally, locally Lipschitz continuous functions possessing certain pointwise-a.e. convexity properties) are Newton differentiable when considered as maps between suitable Lebesgue spaces and equipped with the strong-weak Bouligand differential as a generalized set-valued derivative. It is shown that this Newton differentiability allows to solve optimal control problems with H1-cost terms and one-sided pointwise control constraints by means of a semismooth Newton method. The superlinear convergence of the resulting algorithm is proved in the infinite-dimensional setting and its mesh independence is demonstrated in numerical experiments. We expect that the findings of this paper are also helpful for the design of numerical solution procedures for quasi-variational inequalities and the optimal control of obstacle-type variational problems.

math.OC↗
arXiv2022

Integer optimal control problems with total variation regularization: Optimality conditions and fast solution of subproblems

We investigate local optimality conditions of first and second order for integer optimal control problems with total variation regularization via a finite-dimensional switching point problem. We show the equivalence of local optimality for both problems, which will be used to derive conditions concerning the switching points of the control function. A non-local optimality condition treating back-and-forth switches will be formulated. For the numerical solution, we propose a proximal-gradient method. The emerging discretized subproblems will be solved by employing Bellman's optimality principle, leading to an algorithm which is polynomial in the mesh size and in the admissible control levels. An adaption of this algorithm can be used to handle subproblems of the trust-region method proposed in Leyffer, Manns, 2021. Finally, we demonstrate computational results.

math.OC↗
arXiv2022

Second-order conditions for non-uniformly convex integrands: quadratic growth in $L^1$

We study no-gap second-order optimality conditions for a non-uniformly convex and non-smooth integral functional. The integral functional is extended to the space of measures. The obtained second-order derivatives contain integrals on lower-dimensional manifolds. The proofs utilize the convex pre-conjugate, which is an integral functional on the space of continuous functions. Applications to non-smooth optimal control problems are given.

math.OC↗
arXiv2022

An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints

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.

math.OC↗
arXiv2022

A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces

We give a simple proof of the Baillon-Haddad theorem for convex functions defined on open and convex subsets of Hilbert spaces. We also state some generalizations and limitations. In particular, we discuss equivalent characterizations of the Lipschitz continuity of the derivative of convex functions on open and convex subsets of Banach spaces.

math.FA↗
arXiv2022

Finding global solutions of some inverse optimal control problems using penalization and semismooth Newton methods

We present a method to solve a special class of parameter identification problems for an elliptic optimal control problem to global optimality. The bilevel problem is reformulated via the optimal-value function of the lower-level problem. The reformulated problem is nonconvex and standard regularity conditions like Robinson's CQ are violated. Via a relaxation of the constraints, the problem can be decomposed into a family of convex problems and this is the basis for a solution algorithm. The convergence properties are analyzed. It is shown that a penalty method can be employed to solve this family of problems while maintaining convergence speed. For an example problem, the use of the identity as penalty function allows for the solution by a semismooth Newton method. Numerical results are presented. Difficulties and limitations of our approach to solve a nonconvex problem to global optimality are discussed.

math.OC↗
arXiv2022

From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability

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.

math.OC↗
arXiv2021

Numerical Approximation of Optimal Convex and Rotationally Symmetric Shapes for an Eigenvalue Problem arising in Optimal Insulation

We are interested in the optimization of convex domains under a PDE constraint. Due to the difficulties of approximating convex domains in $\mathbb{R}^3$, 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.

math.OC↗
arXiv2021

M-stationarity for a class of MPCCs in Lebesgue spaces

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.

math.OC↗
arXiv2021

Maximal monotone operators with non-maximal graphical limit

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 $B$.

math.FA↗
arXiv2021

Subdifferentiation of nonconvex sparsity-promoting functionals on Lebesgue spaces

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 $\ell_p$-metric for $p\in[0,1)$ which are nonconvex as well as non-Lipschitz and, thus, variationally challenging. In this paper, we derive exact formulas for the Fréchet, 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.

math.OC↗