Reduced-Order Multiobjective Optimal Control of Semilinear Parabolic Problems

Iapichino, Laura; Trenz, Stefan; Volkwein, Stefan

doi:10.1007/978-3-319-39929-4_37

Cited by 21 publications

(20 citation statements)

References 9 publications

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“…The main difference is now that the objective function may have a more complicated structure. In [105,106], the weighted sum method has been used in combination with RB in order to solve MOPs constrained by elliptic PDEs. In the weighted sum method, scalarization is achieved via convex combination of the individual objectives using the weight vector α:…”

Section: Scalarizationmentioning

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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Section: Scalarizationmentioning

confidence: 99%

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

Peitz

Dellnitz

2018

MCA

View full text Add to dashboard Cite

show abstract

“…The main difference is now that the objective function may have a more complicated structure. In [99,100], the weighted sum method has been used in combination with RB in order to solve MOPs constrained by elliptic PDEs. In the weighted sum method, scalarization is achieved via convex combination of the individual objectives using the weight vector α:…”

Section: Scalarizationmentioning

confidence: 99%

A Survey of Recent Trends in Multiobjective Optimization – Surrogate Models, Feedback Control and Objective Reduction

Peitz¹,

Dellnitz²

2018

Preprint

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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 over recent developments in accelerating multiobjective optimization 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. In the latter case, various promising model predictive control approaches have been proposed. We also briefly comment on dimension reduction in many-objective optimization problems as another technique for reducing the numerical effort.

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“…Let us mention that preliminary results combining reduced-order modeling and multiobjective PDE-constrained optimization have 30 recently been derived; cf. [8,9,13]. The paper is organized in the following manner: In Section 2 we introduce our linear evolution equation as well as our bicriterial optimization problem.…”

Section: Introductionmentioning

confidence: 99%

POD-based error control for reduced-order bicriterial PDE-constrained optimization

Banholzer

Beermann

Volkwein

2017

Annual Reviews in Control

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In the present paper, a bicriterial optimal control problem governed by an abstract evolution problem and bilateral control constraints is considered. To compute Pareto optimal points and the Pareto front numerically, the (Euclidean) reference point method is applied, where many scalar constrained optimization problems have to be solved. For this reason a reduced-order approach based on proper orthogonal decomposition (POD) is utilized. An a-posteriori error analysis ensures a desired accuracy for the Pareto optimal points and for the Pareto front computed by the POD method. Numerical experiments for evolution problems with convection-diffusion illustrate the efficiency of the presented approach.

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Reduced-Order Multiobjective Optimal Control of Semilinear Parabolic Problems

Cited by 21 publications

References 9 publications

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

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

A Survey of Recent Trends in Multiobjective Optimization – Surrogate Models, Feedback Control and Objective Reduction

POD-based error control for reduced-order bicriterial PDE-constrained optimization

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Reduced-Order Multiobjective Optimal Control of Semilinear Parabolic Problems

Cited by 21 publications

References 9 publications

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

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

A Survey of Recent Trends in Multiobjective Optimization&nbsp;&ndash; Surrogate Models, Feedback Control and Objective Reduction

POD-based error control for reduced-order bicriterial PDE-constrained optimization

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A Survey of Recent Trends in Multiobjective Optimization – Surrogate Models, Feedback Control and Objective Reduction