New developments in the primal–dual column generation technique

Abstract

The classical column generation is based on optimal solutions of the restricted master problems. This strategy frequently results in an unstable behaviour and may require an unnecessarily large number of iterations. To overcome this weakness, variations of the classical approach use interior points of the dual feasible set, instead of optimal solutions. In this paper, we address the primal-dual column generation technique, which relies on well-centred non-optimal solutions of the restricted master problems that are obtained by a primal-dual interior point method. Although good computational results are reported for this technique, it was only applied in a particular class of problems. Moreover, no theoretical analysis to guarantee its convergence is available. Here, we further investigate the primaldual column generation technique and present extensive computational experiments in the context of integer programming, where column generation schemes are widely employed. The results show that the primal-dual technique usually leads to substantial reductions in the number of iterations as well as less running time when compared to the classical and also analytic centre approaches.

A new warmstarting strategy for the primal-dual column generation method

“…Termination is achieved when there are no more columns to append and the relative gap drops below a prescribed optimality tolerance. The method has shown encouraging results in solving the root node of several combinatorial optimization problems [15]. Despite the fact the warmstarting strategy presented in this paper was proposed to improve the performance in this specific context, it could be easily extended to any situation where a linear optimization problem is solved using a column generation or cutting plane framework with a primal-dual interior point method.…”

“…In [15], the warmstarting strategy proposed in [14] and further improved by heuristic procedures has been used, showing encouraging results. We will describe this strategy later in the paper because it provides a benchmark to our new warmstarting technique.…”

“…In this paper we will focus on the case when warmstarting is applied in column generation, specifically in the context of the primal-dual column generation method (PDCGM) described in [15]. We address a particularly important application of the column generation method which arises when solving combinatorial optimization problems.…”

“…In this paper we introduce a specialized warmstarting strategy applicable within the primal-dual column generation scheme [15]. Specialized because we take advantage of some structure/properties which are often present when solving combinatorial optimization problems.…”

“…The primal-dual method under which we design our warmstarting strategy is the one described in [15]. This technique relies on well-centred close-tooptimal solutions of the RMPs which are obtained by a primal-dual interior point method.…”