An Inexact Bundle Method and Subgradient Computations for Optimal Control of Deterministic and Stochastic Obstacle Problems

Hertlein, Lukas; Rauls, Anne-Therese; Ulbrich, Michael; Ulbrich, Stefan

doi:10.1007/978-3-030-79393-7_19

Cited by 3 publications

(2 citation statements)

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“…The aim of this paper is to demonstrate that, beside superposition operators and the scalar play and stop considered in [6,7], there is a further large class of operators arising in optimal control applications that are Newton differentiable when endowed with a suitable (and computable) set-valued derivative, namely, solution mappings of obstacle-type variational inequalities (VIs) with unilateral constraints. Such functions arise, for instance, when optimal control problems governed by partial differential equations (PDEs) with H 1 -controls are studied, see sections 5 and 6, or in the field of optimal control of contact problems, see [19,27]. The main idea of our analysis is to exploit that solution maps of obstacle-type VIs possess pointwise-a.e.…”

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confidence: 99%

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

Christof¹,

Wachsmuth²

2021

Preprint

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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 H 1 -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.

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confidence: 99%

“…objective function of, for instance, a tracking-type optimal control problem for the classical obstacle problem is semismooth, it may be possible to use similar techniques in the infinite-dimensional setting, cf. [19,20].…”

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confidence: 99%

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

Christof¹,

Wachsmuth²

2021

Preprint

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

Christof

Wachsmuth

2023

SIAM J. Control Optim.

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Asymptotic Consistency for Nonconvex Risk-Averse Stochastic Optimization with Infinite-Dimensional Decision Spaces

Milz,

Surowiec

2024

Mathematics of OR

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Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these estimators as the sample size goes to infinity. This area of study has a long tradition in stochastic programming. However, the literature is lacking consistency analysis for problems in which the decision variables are taken from an infinite-dimensional space, which arise in optimal control, scientific machine learning, and statistical estimation. By exploiting the typical problem structures found in these applications that give rise to hidden norm compactness properties for solution sets, we prove consistency results for nonconvex risk-averse stochastic optimization problems formulated in infinite-dimensional space. The proof is based on several crucial results from the theory of variational convergence. The theoretical results are demonstrated for several important problem classes arising in the literature.

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An Inexact Bundle Method and Subgradient Computations for Optimal Control of Deterministic and Stochastic Obstacle Problems

Cited by 3 publications

References 35 publications

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

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

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

Asymptotic Consistency for Nonconvex Risk-Averse Stochastic Optimization with Infinite-Dimensional Decision Spaces

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