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A Trust-Region Method for Unconstrained Multiobjective Problems with Applications in Satisficing Processes

  • Kely D. V. Villacorta
  • Paulo R. Oliveira
  • Antoine Soubeyran
Article

Abstract

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.

Keywords

Trust-region methods Unconstrained multiobjective problem Pareto critical point Satisficing process Worthwhile change Variational rationality 

Notes

Acknowledgements

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.

References

  1. 1.
    Ahookhosh, M., Amini, K.: A nonmonotone trust region method with adaptive radius for unconstrained optimization problems. Comput. Math. Appl. 60(3), 411–422 (2010) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bastin, F., Malmedy, V., Mouffe, M., Toint, P.L., Tomanos, D.: A retrospective trust-region method for unconstrained optimization. Math. Program., Ser. A 123(2), 395–418 (2010) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: In: Trust-Region Methods, Philadelphia, PA. MPS-SIAM Series on Optimization (2000) CrossRefGoogle Scholar
  4. 4.
    Erway, J.B., Gill, P.E.: A subspace minimization method for the trust-region step. SIAM J. Optim. 20(3), 1439–1461 (2009) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Gardašević Filipović, M.: A trust region method using subgradient for minimizing a nondifferentiable function. Yugosl. J. Oper. Res. 19(2), 249–262 (2009) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Ji, Y., Li, Y., Zhang, K., Zhang, X.: A new nonmonotone trust-region method of conic model for solving unconstrained optimization. J. Comput. Appl. Math. 233(8), 1746–1754 (2010) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Yu, Z., Zhang, W., Lin, J.: A trust region algorithm with memory for equality constrained optimization. Numer. Funct. Anal. Optim. 29(5–6), 717–734 (2008) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Peng, Y.H., Shi, B.C., Yao, S.B.: Nonmonotone trust region algorithm for linearly constrained multiobjective programming. J. Huazhong Univ. Sci. Technol. Nat. Sci. 31(7), 113–114 (2003) MathSciNetGoogle Scholar
  9. 9.
    Yao, S.B., Shi, B.C., Peng, Y.H.: Nonmonotone trust region algorithms for multiobjective programming with linear constraints. Math. Appl. (Wuhan) 15(suppl.), 55–59 (2002) MathSciNetGoogle Scholar
  10. 10.
    Xi, H., Shi, B.C.: A trust region method for multiobjective programming without constraints. Math. Appl. (Wuhan) 13(3), 67–69 (2000) MATHMathSciNetGoogle Scholar
  11. 11.
    Simon, H.: A behavioral model of rational choice. Q. J. Econ. 69(1), 99–118 (1955) CrossRefGoogle Scholar
  12. 12.
    Soubeyran, A.: Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviors. Mimeo (2009) Google Scholar
  13. 13.
    Soubeyran, A.: Variational rationality and the unsatisfied man: routines and the course pursuit between aspirations, capabilities and beliefs. Mimeo (2010) Google Scholar
  14. 14.
    Attouch, H., Soubeyran, A.: Inertia and reactivity in decision making as cognitive variational inequalities. J. Convex Anal. 13(2), 207–224 (2006) MATHMathSciNetGoogle Scholar
  15. 15.
    Attouch, H., Soubeyran, A.: Local search proximal algorithms as decision dynamics with costs to move. Set-Valued Var. Anal. 19(1), 157–177 (2010) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Martinez-Legaz, J.E., Soubeyran, A.: A tabu search scheme for abstract problems, with applications to the computation of fixed points. J. Math. Anal. Appl. 338(1), 620–627 (2007) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Attouch, H., Redont, P., Soubeyran, A.: A new class of alternating proximal minimization algorithms with costs-to-move. SIAM J. Optim. 18(3), 1061 (2007) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and pde’s. J. Convex Anal. 15(3), 485–506 (2008) MATHMathSciNetGoogle Scholar
  19. 19.
    Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems. An approach based on the Kurdyka- Lojasiewicz inequality. Math. Oper. Res. 35(2), 438–457 (2010) CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Souza, S., Oliviera, P., Cruz Neto, J., Soubeyran, A.: A proximal point algorithm with separable Bregman distances for quasiconvex optimization over the nonnegative orthant. Eur. J. Oper. Res. 201(2), 365–376 (2010) CrossRefMATHGoogle Scholar
  21. 21.
    Luc, T.D., Sarabi, E., Soubeyran, A.: Existence of solutions in variational relations problems without convexity. J. Math. Anal. Appl. 364(2), 544–555 (2010) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Moreno, F., Oliveira, P., Sou Beyran, A.: Proximal algorithm with quasi distance. Application to habits formation. Optimization (2011) Google Scholar
  23. 23.
    Flores-Bazan, F., Luc, T.D., Soubeyran, A.: Maximal elements under reference-dependent preferences with applications to behavioral traps and games. J. Optim. Theory. Appl. 155(3), 883–901 (2012) CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Godal, O., Flam, S., Soubeyran, A.: Gradient differences and bilateral barters. Optimization (2012) Google Scholar
  25. 25.
    Cruz Neto, J.X., Oliveira, P.R., Soaresm, P.A. Jr., Soubeyran, A.: Learning how to play Nash and alternating minimization method for structured nonconvex problems on Riemannian manifolds. J. Convex Anal. 20(2), 395–438 (2013) MATHMathSciNetGoogle Scholar
  26. 26.
    Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51(3), 479–494 (2000) CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Yigui, O., Qian, Z.: A nonmonotonic trust region algorithm for a class of semi-infinite minimax programming. Appl. Math. Comput. 215(2), 474–480 (2009) CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (1999) CrossRefMATHGoogle Scholar
  29. 29.
    Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1995) MATHGoogle Scholar
  30. 30.
    Shi, Z.J., Guo, J.H.: A new trust region method for unconstrained optimization. J. Comput. Appl. Math. 213(2), 509–520 (2008) CrossRefMATHMathSciNetGoogle Scholar
  31. 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
  32. 32.
    Bandura, A., Schunk, D.: Cultivating competence, self efficacy, and intrinsic interest through proximal self motivation. J. Pers. Soc. Psychol. 41, 586–598 (1981) CrossRefGoogle Scholar
  33. 33.
    Brisoux, J.: Le phénomène des ensembles évoqués: une étude empirique des dimensions contenu et taille. Thèse, Université Laval (1995) Google Scholar
  34. 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. 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. 36.
    Oliver, R.: Satisfaction: A Behavioral Perspective on the Consumer. McGraw-Hill, New York (2011) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Kely D. V. Villacorta
    • 1
  • Paulo R. Oliveira
    • 2
    • 3
  • Antoine Soubeyran
    • 4
  1. 1.DCC-CIFederal University of ParaíbaJoão Pessoa-ParaíbaBrazil
  2. 2.PESC-COPPEFederal University of Rio de JaneiroRio de JaneiroBrazil
  3. 3.Centro de TecnologiaCidade UniversitáriaRio de JaneiroBrazil
  4. 4.Aix-Marseille University (Aix-Marseille School of Economics)CNRS & EHESS, Chateau LafargeLes MillesFrance

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