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/s10107020015988
Article, Bibliographie
This is the latest version of this item.
Abstract
The combinatorial integral approximation decomposition splits the optimization of a discretevalued control into two steps: solving a continuous relaxation of the discrete control problem, and computing a discretevalued approximation of the relaxed control. Different algorithms exist for the second step to construct piecewise constant discretevalued 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 controltostate 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/s10107020015988 
Corresponding Links:  
Abstract:  The combinatorial integral approximation decomposition splits the optimization of a discretevalued control into two steps: solving a continuous relaxation of the discrete control problem, and computing a discretevalued approximation of the relaxed control. Different algorithms exist for the second step to construct piecewise constant discretevalued 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 controltostate 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:  Mixedinteger optimal control, Approximation methods, Convergence rates, Combinatorial integral decomposition 
Additional Information:  Special Issue: MixedInteger 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 
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Compactness and convergence rates in the combinatorial integral approximation decomposition. (deposited 23 Apr 2024 12:48)
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