A generic column generation principle: derivation and convergence analysis
Journal article, 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
Variational inequality problems