A note on column generation in Dantzig--Wolfe decomposition algorithms

TLDR: A modification of the column generation operation in Dantzig—Wolfe decomposition is suggested and it is shown how the subproblems may be solved parametrically in such a way as to maximize the immediate improvement in the value of objective in the “master problem”.

Abstract: A modification of the column generation operation in Dantzig--Wolfe decomposition is suggested. Instead of the usual procedure of solving one or more subproblems at each major iteration, it is shown how the subproblems may be solved parametrically in such a way as to maximize the immediate improvement in the value of objective in the "master problem", rather than to maximize the "reduced profit" of the entering column. The parametric problem is shown to involve the maximization of a piece-wise linear concave function of a single variable. It is hoped that in some cases the use of the suggested procedure may improve the slow rates of convergence common in decomposition algorithms.