Using column generation to compute lower bound sets for bi-objective combinatorial optimization problems

¹ CNRS, LAAS, 7 avenue du colonel Roche, 31400 Toulouse, France.
bmsarpon@laas.fr
² University de Toulouse, INSA, LAAS, 31400 Toulouse, France.
artigues@laas.fr
³ University de Toulouse, LAAS, 31400 Toulouse, France.
njozefow@laas.fr

Received: 29 September 2014
Accepted: 16 October 2014

Abstract

We discuss the use of column generation in a bi-objective setting. Just as in single objective combinatorial optimization, the role of column generation in the bi-objective setting is to compute dual bounds (i.e. lower bounds for minimization problems and upper bounds for maximization problems) which can be used to guide the search for efficient solutions or to evaluate the quality of approximate solutions. The general idea used in this paper is to first transform the bi-objective problem into single objective by a scalarization method and then solve the transformed problem several times by varying the necessary parameters. We show that irrespective of the scalarization method used, similar subproblems are solved when applying column generation. For this reason, we investigate possible ways of intelligently searching for columns for these subproblems in order to accelerate the column generation method.