International Journal of Pure and Applied Mathematics ————————————————————————– Volume 47 No. 4 2008, 501-518 ITERATIVE REFINEMENT FOR INVARIANT SUBSPACES OF MATRICES WITH APPLICATION TO THE PROMETHEE-GAIA METHOD Mario Ahues 1 § , David Duvivier 2 , Alain Largillier 3 , Nadine Meskens 4 1,3 Laboratoire de Math´ ematiques de l’Universit´ e de Saint-Etienne (LaMUSE) Universit´ e Jean Monnet 23 Rue du Dr. Paul Michelon Saint- ´ Etienne, 42100, FRANCE 1 e-mail: mario.ahues@univ-st-etienne.fr 3 e-mail: larg@univ-st-etienne.fr 2 Laboratoire d’Informatique du Littoral Universit´ e du Littoral Cˆ ote d’Opale Maison de la Recherche Blaise Pascal 50, Rue Ferdinand Buisson - BP 719, Calais, 62228, FRANCE e-mail: david.duvivier@lil.univ-littoral.fr 4 Louvain School of Management and Catholic University of Mons (FUCaM) 151, Chauss´ ee de Binche, Mons, 7000, BELGIUM e-mail: nadine.meskens@fucam.ac.be Abstract: Recent results in operational research have provided decision mak- ers with an adaptable tool that is able to synthesize the performance measures of several solutions to be compared in a short time. In the context of multi- criteria limited-time decision making problems, two of the authors have devel- oped hybrid models composed of two mathematical models, a set of dedicated heuristics, a stochastic local search, meta-heuristic and a simulation model. According to the decision makers, the solutions are ranked on the basis of sev- Received: August 18, 2008 c  2008, Academic Publications Ltd. § Correspondence author