Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions

A state-constrained optimal control problem with nonlocal radiation interface conditions arising from the modeling of crystal growth processes is considered. The problem is approximated by a Moreau-Yosida type regularization. Optimality conditions for the regularized problem are derived and the convergence of the regularized problems is shown. In the last part of the paper, some numerical results are presented

Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions

A state-constrained optimal control problem with nonlocal radiation interface conditions arising from the modeling of crystal growth processes is considered. The problem is approximated by a Moreau-Yosida type regularization. Optimality conditions for the regularized problem are derived and the convergence of the regularized problems is shown. In the last part of the paper, some numerical results are presented

Forward-backward truncated Newton methods for convex composite optimization

This paper proposes two proximal Newton-CG methods for convex nonsmooth optimization problems in composite form. The algorithms are based on a a reformulation of the original nonsmooth problem as the unconstrained minimization of a continuously differentiable function, namely the forward-backward envelope (FBE). The first algorithm is based on a standard line search strategy, whereas the second one combines the global efficiency estimates of the corresponding first-order methods, while achieving fast asymptotic convergence rates. Furthermore, they are computationally attractive since each Newton iteration requires the approximate solution of a linear system of usually small dimension

Total variation regularization of multi-material topology optimization

This work is concerned with the determination of the diffusion coefficient from distributed data of the state. This problem is related to homogenization theory on the one hand and to regularization theory on the other hand. An approach is proposed which involves total variation regularization combined with a suitably chosen cost functional that promotes the diffusion coefficient assuming prespecified values at each point of the domain. The main difficulty lies in the delicate functional-analytic structure of the resulting nondifferentiable optimization problem with pointwise constraints for functions of bounded variation, which makes the derivation of useful pointwise optimality conditions challenging. To cope with this difficulty, a novel reparametrization technique is introduced. Numerical examples using a regularized semismooth Newton method illustrate the structure of the obtained diffusion coefficient.

Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques

Solution of elliptic optimal control problem with pointwise and non-local state constraints

© 2017, Allerton Press, Inc.We study an optimal control problem of a system governed by a linear elliptic equation, with pointwise control constraints and pointwise and non-local (integral) state constraints. We construct a finite-difference approximation of the problem, we prove the existence and the convergence of the approximate solutions to the exact solution. We construct and study mesh saddle point problem and its iterative solution method and analyze the results of numerical experiments

Numerische Konzepte und Fehleranalysis zu elliptischen Randsteuerungsproblemen mit punktweisen Zustands- und Kontrollbeschränkungen

Optimization in technical applications described by partial differential equations plays a more and more important role. By means of the control the solution of a partial differential equation called state is influenced. Simultaneously a cost functional has to be minimized. In many technical applications pointwise constraints to the state or the control are reasonable. It is well known that the Lagrange multipliers with respect to pure state constraints are in general only regular Borel measures. This fact implies a lower regularity of the optimal solution of the problem. In this dissertation a linear quadratic optimal control problem governed by an elliptic partial differential equation an Neumann boundary control is investigated. Furthermore, we consider pointwise state constraints in an inner subdomain and bilateral constraints on the boundary control. Despite the above mentioned problems, we benefit from the localization of the Lagrange multiplier in the inner subdomain such that a higher regularity of the optimal control is shown. However, the so called dual variables of the optimal control problem are not unique. Hence, the application of well known and efficient optimization algorithms becomes difficult. Presenting a regularization concept, we will avoid these problems. We introduce an additional distributed control ("virtual control") which appears in the cost functional, the right hand side of the partial differential equation and in the regularized state constraints. The effect of regularization is influenced by several parameter functions. We derive an error estimate for the error between the optimal solution of the original problem and the regularized one. Moreover, under some assumptions on the parameter functions we obtain certain convergence rates of the regularization error. In the following a finite element based approximation of the regularized optimal control problems is established. Based on appropriate feasible test functions, we derive an error estimate between the optimal solution of the unregularized original problem and the regularized and discretized one. Thereby, we consider the regularization and discretization simultaneously and we propose a suitable coupling of the parameter functions and the mesh size. Forthcoming, we present the primal-dual active set strategy as a optimization method for solving the regularized optimal control problems. Moreover, we derive an error estimate between the current iterates of the algorithm and the optimal solution. Based on this, we construct an error estimator, which is reliable as an alternative stopping criterion for the primal-dual active set strategy. Finally, the theoretical results of this work are illustrated by several numerical examples.Physikalische und technische Anwendungen werden häufig durch partielle Differentialgleichungen beschrieben. Die Optimierung solcher Prozesse führt auf sogenannte Optimalsteuerprobleme mit partiellen Differentialgleichungen. Mit Hilfe einer Steuerungsvariable wird die Lösung der Differentialgleichung, welche Zustand genannt wird, beeinflusst. Gleichzeitig soll ein Zielfunktional minimiert werden. Bei vielen technischen Anwendungen sind punktweise Beschränkungen an den Zustand oder die Steuerung sinnvoll. Es ist bekannt, dass die zu den Zustandsbeschränkungen gehörigen Lagrangsche Multiplikatoren im allgemeinen nur reguläre Borel-Maße sind. Dies führt zu einer geringeren Regularität der optimalen Lösung des Problems. In dieser Dissertationsschrift wird ein linear-quadratisches Optimalsteuerproblem mit elliptischer partieller Differentialgleichung und Neumann-Randsteuerung untersucht. Wir betrachten punkteweise Zustandsschranken in einem inneren Teilgebiet und bilaterale Schranken an die Randsteuerung. Die räumliche Trennung der Zustandsbeschränkungen von dem Wirkungsgebiet der Steuerung gestattet an vielen Stellen den Einsatz von speziell konstruierten mathematischen Techniken. Dies betrifft sowohl Regularitätsaussagen als auch Fehlerabschätzungen. Allerdings sind die sogenannten dualen Variablen des Problems nicht eindeutig. Dies macht die Anwendung bekannter effizienter Optimierungsalgorithmen unmöglich. Es wird ein Regularisierungskonzept vorgestellt, um dieses Problem zu vermeiden. Dabei wird eine zusätzliche verteilte Steuerung ("virtuelle Steuerung") eingeführt, welche im Zielfunktional, in der rechten Seite der Differentialgleichungen und in den regularisierten Zustandsbeschränkungen auftaucht. Die Regularisierung wird durch verschiedene Parameterfunktionen beeinflusst. Wir leiten Abschätzungen für den Fehler zwischen der optimalen Lösung des Ausgangsproblems und der des regularisierten Problems her. Bei Verwendung geschickt gewählter Parameterfunktionen ergeben sich aus diesen Abschätzungen direkt Konvergenzraten für die Regularisierung. Im weiteren betrachten wir auch eine Diskretisierung des regularisierten Problems mit Hilfe von finiten Elementen. Basierend auf geeignet konstruierten zulässigen Testfunktionen wird eine Fehlerabschätzung der optimalen Lösung des unregularisierten Problems zur diskretisierten und regularisierten Lösung hergeleitet. Da der Regularisierungs- und der Diskretisierungsfehler gleichzeitig auftreten, wird eine geeignete Kopplung des Regularisierungsparameters mit der Gitterweite angegeben. Eine primal-duale aktive Mengenstrategie wird als Optimierungsalgorithmus zur Lösung der regularisierten Probleme vorgestellt. Weiterhin wird eine Fehlerabschätzung der aktuellen Iterierten dieses Algorithmus zur optimalen Lösung bewiesen. Basierend auf diesem Resultat wird ein Fehlerschätzer konstruiert, welcher als alternatives Abbruchkriterium für die aktive Mengenstrategie benutzt werden kann. Die Resultate der Arbeit werden durch verschiedene numerische Beispiele bestätigt

Preconditioning of Active-Set Newton Methods for PDE-constrained Optimal Control Problems

We address the problem of preconditioning a sequence of saddle point linear systems arising in the solution of PDE-constrained optimal control problems via active-set Newton methods, with control and (regularized) state constraints. We present two new preconditioners based on a full block matrix factorization of the Schur complement of the Jacobian matrices, where the active-set blocks are merged into the constraint blocks. We discuss the robustness of the new preconditioners with respect to the parameters of the continuous and discrete problems. Numerical experiments on 3D problems are presented, including comparisons with existing approaches based on preconditioned conjugate gradients in a nonstandard inner product