A generic column generation principle: derivation and convergence analysis
Artikel i vetenskaplig tidskrift, 2015

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.

Sequential linear programming

Column generation

Simplicial decomposition

Variational inequality problems

Saddle-point problems

Convex programming

Dantzig-Wolfe decomposition

Författare

Torbjörn Larsson

Linköpings universitet

Athanasios Migdalas

Luleå tekniska universitet

Michael Patriksson

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

Operational Research

1109-2858 (ISSN) 1866-1505 (eISSN)

Vol. 15 163-198

Styrkeområden

Informations- och kommunikationsteknik

Transport

Produktion

Energi

Drivkrafter

Hållbar utveckling

Ämneskategorier

Beräkningsmatematik

Fundament

Grundläggande vetenskaper

DOI

10.1007/s12351-015-0171-3