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A generic column generation principle: derivation and convergence analysis

Journal article
Authors Torbjörn Larsson
Athanasios Migdalas
Michael Patriksson
Published in Operational Research: An International Journal
Volume 15
Issue 2
Pages 163-198
ISSN 1109-2858
Publication year 2015
Published at Department of Mathematical Sciences, Mathematics
Pages 163-198
Language en
Links dx.doi.org/10.1007/s12351-015-0171-...
Keywords Convex programming; Variational inequality problems; Saddle-point problems; Column generation; Simplicial decomposition; Dantzig-Wolfe decomposition; Sequential linear programming
Subject categories Numerical analysis, Optimization, systems theory

Abstract

Given a non-empty, compact and convex set, and an a priori defined condition which each element either satisfies or not, we want to find an element belonging to the former category. This is a fundamental problem of mathematical programming which encompasses nonlinear programs, variational inequalities, and saddle-point problems. We present a conceptual column generation scheme, which alternates between solving a restriction of the original problem and a column generation phase which is used to augment the restricted problems. We establish the general applicability of the conceptual method, as well as to the three problem classes mentioned. We also establish a version of the conceptual method in which the restricted and column generation problems are allowed to be solved approximately, and of a version allowing for the dropping of columns. We show that some solution methods (e.g., Dantzig-Wolfe decomposition and simplicial decomposition) are special instances, and present new convergent column generation methods in nonlinear programming, such as a sequential linear programming (SLP) type method. Along the way, we also relate our quite general scheme in nonlinear programming presented in this paper with several other classic, and more recent, iterative methods in nonlinear optimization.

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