A Trust-Region Method for Unconstrained Multiobjective Problems with Applications in Satisficing Processes
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Multiobjective optimization has a significant number of real-life applications. For this reason, in this paper we consider the problem of finding Pareto critical points for unconstrained multiobjective problems and present a trust-region method to solve it. Under certain assumptions, which are derived in a very natural way from assumptions used to establish convergence results of the scalar trust-region method, we prove that our trust-region method generates a sequence which converges in the Pareto critical way. This means that our generalized marginal function, which generalizes the norm of the gradient for the multiobjective case, converges to zero. In the last section of this paper, we give an application to satisficing processes in Behavioral Sciences. Multiobjective trust-region methods appear to be remarkable specimens of much more abstract satisficing processes, based on “variational rationality” concepts. One of their important merits is to allow for efficient computations. This is a striking result in Behavioral Sciences.
KeywordsTrust-region methods Unconstrained multiobjective problem Pareto critical point Satisficing process Worthwhile change Variational rationality
This work was partially supported by CNPq—Brasil. The authors wish to thank the anonymous referees for carefully reading the paper and providing valuable comments and suggestions which helped them improve the presentation.
- 12.Soubeyran, A.: Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviors. Mimeo (2009) Google Scholar
- 13.Soubeyran, A.: Variational rationality and the unsatisfied man: routines and the course pursuit between aspirations, capabilities and beliefs. Mimeo (2010) Google Scholar
- 22.Moreno, F., Oliveira, P., Sou Beyran, A.: Proximal algorithm with quasi distance. Application to habits formation. Optimization (2011) Google Scholar
- 24.Godal, O., Flam, S., Soubeyran, A.: Gradient differences and bilateral barters. Optimization (2012) Google Scholar
- 31.Lewin, K., Dembo, T., Festinger, L., Sears, P.: Level of aspiration. In: Personality and the Behavior Disorders. Ronald Press, New York (1994) Google Scholar
- 33.Brisoux, J.: Le phénomène des ensembles évoqués: une étude empirique des dimensions contenu et taille. Thèse, Université Laval (1995) Google Scholar
- 34.Brisoux, J., Laroche, M.: Evoked set formation and composition: an empirical investigation under a routinized response behavior situation. In: Monroe, K. (ed.) Advances in Consumer Research, vol. 8, pp. 357–361. Ann Arbor, Michigan (1981). Association for Consumer Research Google Scholar
- 35.Jolivot, A.: Thirty years of research on consideration set: a state of the art. Série “Recherche”. W.P. 502, Institut d’Administration des Entreprises, Clos Guiot p. 13540 Puyricard, France (1997) Google Scholar
- 36.Oliver, R.: Satisfaction: A Behavioral Perspective on the Consumer. McGraw-Hill, New York (2011) Google Scholar