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

Jacek Gondzio (j.gondzioed.ac.uk)
Pablo González-Brevis (pablogonzalezingenieros.udd.cl)
Pedro Munari (munaridep.ufscar.br)

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. A reduction 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. In this paper, we investigate the behaviour of the PDCGM 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 state-of-the-art methods. The results suggest that the PDCGM offers an attractive alternative over specialized methods since it remains competitive in terms of number of iterations and CPU times.

Keywords: column generation, cutting plane method, interior point methods, convex optimization, multiple kernel learning problem, two-stage stochastic programming, multicommodity network flow problem

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Linear, Cone and Semidefinite Programming (Linear Programming )

Category 3: Optimization Software and Modeling Systems

Citation: Technical Report, ERGO 13-014, University of Edinburgh, School of Mathematics (2013)

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Entry Submitted: 09/09/2013
Entry Accepted: 09/09/2013
Entry Last Modified: 01/22/2015

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