Set-Oriented Multiobjective Optimal Control of PDEs Using Proper Orthogonal Decomposition

Beermann, Dennis; Dellnitz, Michael; Peitz, Sebastian; Volkwein, Stefan

doi:10.1007/978-3-319-75319-5_3

Cited by 15 publications

(15 citation statements)

References 17 publications

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“…For future work, it may be beneficial to investigate where this drastic overestimation comes from and how it can be placed in check. In [3], a heuristic was deployed in which real error and estimator for some random points were calculated offline before the optimization in order to and gauge the overestimation factor.…”

Section: Discussionmentioning

confidence: 99%

POD-Based Mixed-Integer Optimal Control of the Heat Equation

Bachmann

Beermann

et al. 2019

J Sci Comput

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In the present paper an optimal control problem governed by the heat equation is considered, where continuous as well as discrete controls are involved. To obtain the discrete controls the branch-and-bound method is utilized, where in each node a relaxed control constrained optimal control problem has to be solved involving only continuous controls. However, the solutions to many relaxed optimal control problems have to be computed numerically. For that reason model-order reduction is applied to speed-up the branch-and-bound method. In this work the method of proper orthogonal decomposition (POD) is used. A posteriori error estimation in each node ensures that the calculated solutions are sufficiently accurate. Numerical experiments illustrate the efficiency of the proposed strategy.

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Section: Discussionmentioning

confidence: 99%

POD-Based Mixed-Integer Optimal Control of the Heat Equation

Bachmann

Beermann

et al. 2019

J Sci Comput

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“…This scenario occurs frequently for complex physics simulations, e.g., when the system under consideration is described by a partial differential equation, cf. [23,2,26,3] for examples. Here, while it is often possible to calculate single Pareto critical points, the computation of the full Pareto critical set via a fine pointwise approximation is computationally infeasible.…”

Section: Application 2: Generation Of Surrogate Models Of Expensive Mopsmentioning

confidence: 99%

Inverse multiobjective optimization: Inferring decision criteria from data

Gebken,

Peitz

2019

Preprint

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It is a very challenging task to identify the objectives on which a certain decision was based, in particular if several, potentially conflicting criteria are equally important and a continuous set of optimal compromise decisions exists. This task can be understood as the inverse problem of multiobjective optimization, where the goal is to find the objective vector of a given Pareto set. To this end, we present a method to construct the objective vector of a multiobjective optimization problem (MOP) such that the Pareto critical set contains a given set of data points or decision vectors. The key idea is to consider the objective vector in the multiobjective KKT conditions as variable and then search for the objectives that minimize the Euclidean norm of the resulting system of equations. By expressing the objectives in a finite-dimensional basis, we transform this problem into a homogeneous, linear system of equations that can be solved efficiently. There are many important potential applications of this approach. Besides the identification of objectives (both from clean and noisy data), the method can be used for the construction of surrogate models for expensive MOPs, which yields significant speed-ups. Both applications are illustrated using several examples.

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“…First results coupling the subdivision algorithm developed in [27] with error estimates for POD-based ROMs have recently appeared [110,111]. In the subdivision algorithm, the decision space is divided into boxes, which are alternatingly subdivided and selected.…”

Section: Set-oriented Approaches With -Dominancementioning

confidence: 99%

A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction

Peitz

Dellnitz

2018

MCA

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Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control, which results in new challenges such as expensive models or real-time applicability. Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in accelerating multiobjective optimal control for complex problems where either PDE constraints are present or where a feedback behavior has to be achieved. In the first case, surrogate models yield significant speed-ups. Besides classical meta-modeling techniques for multiobjective optimization, a promising alternative for control problems is to introduce a surrogate model for the system dynamics. In the case of real-time requirements, various promising model predictive control approaches have been proposed, using either fast online solvers or offline-online decomposition. We also briefly comment on dimension reduction in many-objective optimization problems as another technique for reducing the numerical effort.

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Set-Oriented Multiobjective Optimal Control of PDEs Using Proper Orthogonal Decomposition

Cited by 15 publications

References 17 publications

POD-Based Mixed-Integer Optimal Control of the Heat Equation

POD-Based Mixed-Integer Optimal Control of the Heat Equation

Inverse multiobjective optimization: Inferring decision criteria from data

A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction

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