Stabilization in Column Generation

Column Generation (CG) algorithms are instrumental in many areas of applied optimization, where Linear Programs with an enormous number of columns need to be solved. Although successfully used in many applications, the standard CG algorithm suffers from well-known “instability” issues that somewhat limit its efficiency and usability. Building on the theory developed for NonDifferentiable Optimization algorithm, we propose a large class of Stabilized Column Generation (SCG) algorithms which avoid the instability problems of the standard approach by using an explicit stabilizing term in the dual; this amounts at considering a (generalized) Augmented Lagrangian of the primal Master Problem. Since the theory allows a great degree of flexibility in the choice and in the management of the stabilizing term, we can use piecewise-linear functions that can be efficiently dealt with off-the-shelf LP technology, as well as being related in interesting ways with some previous attempt at stabilizing the CG algorithm. The effectiveness in practice of this approach is demonstrated by extensive computational experiments on large-scale Multi-Depot Vehicle Scheduling problems and simultaneous Vehicle and Crew Scheduling problems.