TU Darmstadt / ULB / TUbiblio

Compactness and convergence rates in the combinatorial integral approximation decomposition

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
Article, Bibliographie

This is the latest version of this item.

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.

Item Type: Article
Erschienen: 2021
Creators: Kirches, Christian ; Manns, Paul ; Ulbrich, Stefan
Type of entry: Bibliographie
Title: Compactness and convergence rates in the combinatorial integral approximation decomposition
Language: English
Date: August 2021
Place of Publication: Berlin ; Heidelberg
Publisher: Springer
Journal or Publication Title: Mathematical Programming: Series A, Series B
Volume of the journal: 188
Issue Number: 2
DOI: 10.1007/s10107-020-01598-8
Corresponding Links:
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.

Uncontrolled Keywords: Mixed-integer optimal control, Approximation methods, Convergence rates, Combinatorial integral decomposition
Additional Information:

Special Issue: Mixed-Integer Nonlinear Programming in Oberwolfach

Mathematics Subject Classification: 41A25 · 49M20 · 49M25 · 65D15 · 90C11

Classification DDC: 500 Science and mathematics > 510 Mathematics
Divisions: 04 Department of Mathematics
04 Department of Mathematics > Optimization
Date Deposited: 25 Apr 2024 10:16
Last Modified: 25 Apr 2024 10:16
PPN:
Corresponding Links:
Export:
Suche nach Titel in: TUfind oder in Google

Available Versions of this Item

Send an inquiry Send an inquiry

Options (only for editors)
Show editorial Details Show editorial Details