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A generic column generation principle: derivation and convergence analysis

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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 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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Notes

  1. The requirement that f be convex is only made for reasons of simplicity of the presentation.

  2. As  Higgins and Polak (1990) have pointed out, von Hohenbalken’s version of the algorithm is guaranteed convergent for polyhedral feasible sets only.

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Authors and Affiliations

  1. Department of Mathematics, Linköping University, 581 83, Linköping, Sweden

    Torbjörn Larsson

  2. Department of Business Administration, Technology and Social Sciences, Luleå University of Technology, 971 87, Luleå, Sweden

    Athanasios Migdalas

  3. Department of Mathematical Sciences, Chalmers University of Technology, 412 96, Gothenburg, Sweden

    Michael Patriksson

Authors
  1. Torbjörn Larsson
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  2. Athanasios Migdalas
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  3. Michael Patriksson
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Correspondence to Athanasios Migdalas.

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Larsson, T., Migdalas, A. & Patriksson, M. A generic column generation principle: derivation and convergence analysis. Oper Res Int J 15, 163–198 (2015). https://doi.org/10.1007/s12351-015-0171-3

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