Kirches, Christian ; Manns, Paul ; Ulbrich, Stefan (2021)
Compactness and convergence rates in the combinatorial integral approximation decomposition.
In: Mathematical Programming: Series A, Series B, 188 (2)
doi: 10.1007/s10107-020-01598-8
Artikel, Bibliographie
Dies ist die neueste Version dieses Eintrags.
Kurzbeschreibung (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.
| Typ des Eintrags: | Artikel |
|---|---|
| Erschienen: | 2021 |
| Autor(en): | Kirches, Christian ; Manns, Paul ; Ulbrich, Stefan |
| Art des Eintrags: | Bibliographie |
| Titel: | Compactness and convergence rates in the combinatorial integral approximation decomposition |
| Sprache: | Englisch |
| Publikationsjahr: | August 2021 |
| Ort: | Berlin ; Heidelberg |
| Verlag: | Springer |
| Titel der Zeitschrift, Zeitung oder Schriftenreihe: | Mathematical Programming: Series A, Series B |
| Jahrgang/Volume einer Zeitschrift: | 188 |
| (Heft-)Nummer: | 2 |
| DOI: | 10.1007/s10107-020-01598-8 |
| Zugehörige Links: | |
| Kurzbeschreibung (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. |
| Freie Schlagworte: | Mixed-integer optimal control, Approximation methods, Convergence rates, Combinatorial integral decomposition |
| Zusätzliche Informationen: | Special Issue: Mixed-Integer Nonlinear Programming in Oberwolfach Mathematics Subject Classification: 41A25 · 49M20 · 49M25 · 65D15 · 90C11 |
| Sachgruppe der Dewey Dezimalklassifikatin (DDC): | 500 Naturwissenschaften und Mathematik > 510 Mathematik |
| Fachbereich(e)/-gebiet(e): | 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Optimierung |
| Hinterlegungsdatum: | 25 Apr 2024 10:16 |
| Letzte Änderung: | 25 Apr 2024 10:16 |
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Compactness and convergence rates in the combinatorial integral approximation decomposition. (deposited 23 Apr 2024 12:48)
- Compactness and convergence rates in the combinatorial integral approximation decomposition. (deposited 25 Apr 2024 10:16) [Gegenwärtig angezeigt]
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