Large-scale optimization with the primal-dual column generation method

The primal-dual column generation method (PDCGM) is a general-purpose column generation technique that relies on the primal-dual interior point method to solve the restricted master problems. The use of this interior point method variant allows to obtain suboptimal and well-centered dual solutions w...

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Published in:Mathematical programming computation Vol. 8; no. 1; pp. 47 - 82
Main Authors: Gondzio, Jacek, González-Brevis, Pablo, Munari, Pedro
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2016
Subjects:
Full Length Paper
Mathematics
Mathematics and Statistics
Mathematics of Computing
Operations Research/Decision Theory
Optimization
Theory of Computation
90C25 Convex programming
90C51 Interior-point methods
Column generation
Multiple kernel learning problem
49M27 Decomposition methods
90C06 Large-scale problems
Convex optimization
Two-stage stochastic programming
Interior point methods
Multicommodity network flow problem
Cutting plane method
ISSN:1867-2949, 1867-2957
Online Access:Get full text
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Abstract The primal-dual column generation method (PDCGM) is a general-purpose column generation technique that relies on the primal-dual interior point method to solve the restricted master problems. The use of this interior point method variant allows to obtain suboptimal and well-centered dual solutions which naturally stabilizes the column generation process. As recently presented in the literature, reductions in the number of calls to the oracle and in the CPU times are typically observed when compared to the standard column generation, which relies on extreme optimal dual solutions. However, these results are based on relatively small problems obtained from linear relaxations of combinatorial applications. In this paper, we investigate the behaviour of the PDCGM in a broader context, namely when solving large-scale convex optimization problems. We have selected applications that arise in important real-life contexts such as data analysis (multiple kernel learning problem), decision-making under uncertainty (two-stage stochastic programming problems) and telecommunication and transportation networks (multicommodity network flow problem). In the numerical experiments, we use publicly available benchmark instances to compare the performance of the PDCGM against recent results for different methods presented in the literature, which were the best available results to date. The analysis of these results suggests that the PDCGM offers an attractive alternative over specialized methods since it remains competitive in terms of number of iterations and CPU times even for large-scale optimization problems.
AbstractList The primal-dual column generation method (PDCGM) is a general-purpose column generation technique that relies on the primal-dual interior point method to solve the restricted master problems. The use of this interior point method variant allows to obtain suboptimal and well-centered dual solutions which naturally stabilizes the column generation process. As recently presented in the literature, reductions in the number of calls to the oracle and in the CPU times are typically observed when compared to the standard column generation, which relies on extreme optimal dual solutions. However, these results are based on relatively small problems obtained from linear relaxations of combinatorial applications. In this paper, we investigate the behaviour of the PDCGM in a broader context, namely when solving large-scale convex optimization problems. We have selected applications that arise in important real-life contexts such as data analysis (multiple kernel learning problem), decision-making under uncertainty (two-stage stochastic programming problems) and telecommunication and transportation networks (multicommodity network flow problem). In the numerical experiments, we use publicly available benchmark instances to compare the performance of the PDCGM against recent results for different methods presented in the literature, which were the best available results to date. The analysis of these results suggests that the PDCGM offers an attractive alternative over specialized methods since it remains competitive in terms of number of iterations and CPU times even for large-scale optimization problems.
Author Munari, Pedro
Gondzio, Jacek
González-Brevis, Pablo
Author_xml – sequence: 1
  givenname: Jacek
  surname: Gondzio
  fullname: Gondzio, Jacek
  organization: School of Mathematics, The University of Edinburgh
– sequence: 2
  givenname: Pablo
  surname: González-Brevis
  fullname: González-Brevis, Pablo
  organization: School of Engineering, Universidad del Desarrollo
– sequence: 3
  givenname: Pedro
  surname: Munari
  fullname: Munari, Pedro
  email: munari@dep.ufscar.br
  organization: Production Engineering Department, Federal University of São Carlos
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Issue 1
Keywords 90C25 Convex programming
90C51 Interior-point methods
Column generation
Multiple kernel learning problem
49M27 Decomposition methods
90C06 Large-scale problems
Convex optimization
Two-stage stochastic programming
Interior point methods
Multicommodity network flow problem
Cutting plane method
Language English
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Snippet The primal-dual column generation method (PDCGM) is a general-purpose column generation technique that relies on the primal-dual interior point method to solve...
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SubjectTerms Full Length Paper
Mathematics
Mathematics and Statistics
Mathematics of Computing
Operations Research/Decision Theory
Optimization
Theory of Computation
Title Large-scale optimization with the primal-dual column generation method
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