Compactness and convergence rates in the combinatorial integral approximation decomposition
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Compactness and convergence rates in the combinatorial integral approximation decomposition Christian Kirches1 · Paul Manns1
· Stefan Ulbrich2
Received: 29 January 2020 / Accepted: 13 November 2020 © The Author(s) 2020
Abstract The combinatorial integral approximation decomposition splits the optimization of a discrete-valued control into two steps: solving a continuous relaxation of the discrete control problem, and computing a discrete-valued approximation of the relaxed control. Different algorithms exist for the second step to construct piecewise constant discrete-valued approximants that are defined on given decompositions of the domain. It is known that the resulting discrete controls can be constructed such that they converge to a relaxed control in the weak∗ topology of L ∞ if the grid constant of this decomposition is driven to zero. We exploit this insight to formulate a general approximation result for optimization problems, which feature discrete and distributed optimization variables, and which are governed by a compact control-to-state operator. We analyze the topology induced by the grid refinements and prove convergence rates of the control vectors for two problem classes. We use a reconstruction problem from signal processing to demonstrate both the applicability of the method outside the scope of differential equations, the predominant case in the literature, and the effectiveness of the approach. Keywords Mixed-integer optimal control · Approximation methods · Convergence rates · Combinatorial integral decomposition Mathematics Subject Classification 41A25 · 49M20 · 49M25 · 65D15 · 90C11
C. Kirches and P. Manns have been supported by the Deutsche Forschungsgemeinschaft (DFG) through Priority Programme 1962, Project 15. Stefan Ulbrich has been supported by the DFG within TRR 154 Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks, subprojects A01 and A02.
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Paul Manns [email protected]
1
Technische Universität Braunschweig, Universitätsplatz 2, 38106 Brunswick, Germany
2
Technische Universität Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany
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1 Introduction This article concerns the following class of optimization problems inf j(K (x)) x
s.t.x ∈ L ∞ (ΩT , V ), x(s) ∈ {ξ1 , . . . , ξ M } ⊂ V for almost all (a.a.) s ∈ ΩT
(P)
for some bounded domain ΩT ⊂ Rd , d ≥ 1. It is an infinite-dimensional and nonsmooth optimization problem, in which the distributed optimization variable x is restricted to a finite number of realizations, often also called bangs. The control-tostate operator K solves the dynamics of the underlying system that is controlled. We note that the feasible set of (P) is bounded. However, we cannot generally expect that (P) admits a minimizer because the feasible set of (P) is not closed in the weak∗ topology. Apart from control of ODEs and PDEs with discrete-valued control inputs, the problem class (P) also covers problems from image denoising and topology optimization. However, we note that there ar
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