Column Generation Algorithms for Nonlinear Optimization II: Numerical Investigations

Resumen

García et al. present a class of column generation (CG) algorithms for nonlinear programs. Its main
motivation from a theoretical viewpoint is that under some circumstances, finite convergence can be
achieved, in much the same way as for the classic simplicial decomposition method; the main practical
motivation is that within the class there are certain nonlinear column generation problems that can
accelerate the convergence of a solution approach which generates a sequence of feasible points. This
algorithm can, for example, accelerate simplicial decomposition schemes by making the subproblems
nonlinear. This paper complements the theoretical study on the asymptotic and finite convergence of
these methods given in
[1]
with an experimental study focused on their computational efficiency.
Three types of numerical experiments are conducted. The first group of test problems has been
designed to study the parameters involved in these methods. The second group has been designed to
investigate the role and the computation of the prolongation of the generated columns to the relative
boundary. The last one has been designed to carry out a more complete investigation of the difference
in computational efficiency between linear and nonlinear column generation approaches.
In order to carry out this investigation, we consider two types of test problems: the first one is the
nonlinear, capacitated single-commodity network flow problem of which several large-scale instances
with varied degrees of nonlinearity and total capacity are constructed and investigated, and the second
one is a combined traffic assignment model