Large-Scale Optimization with the Primal-Dual Column Generation Method

Technical Report ERGO-13-014

J. Gondzio, P. P. González-Brevis, P. Munari

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 stabilize 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.

Key words: Column Generation, Cutting Plane Method, Interior Point Methods, Convex Optimization, Multiple Kernel Learning Problem, Two-stage Stochastic Programming, Multicommodity Network Flow Problem

PDF ERGO-13-014.pdf.

Written: September 3, 2013, revised January 22, 2015.
Mathematical Programming Computation. Vol 8 (2016) pp. 47--82.
Published online September 3, 2015.

Related Software:
PDCGM Primal-Dual Column Generation Method.