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

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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.

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Notes

  1. “bck” stands for “backtraking”.

  2. “ubs” stands for “upper bound on the slope”.

  3. “apx” stands for “approximate”.

  4. “dap” stands for “decrease at the approximate point”.

  5. “dla m” stands for “decrease local approximation m k ”.

  6. “uFh” stands for “upper bound on the \(\underline{F}\)’s Hessian”.

  7. “ umh” stands for “upper bound on the \(\underline{m_{k}}\)’s Hessian”.

  8. “ubh” stands for “upper bound on the Hessians”.

  9. “lb ω” stands for “lower bound on \(\underline{\omega }\)”.

  10. “lb Δ” stands for “lower bound on Δ”.

References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  3. Conn, A.R., Gould, N.I.M., Toint, P.L.: In: Trust-Region Methods, Philadelphia, PA. MPS-SIAM Series on Optimization (2000)

    Chapter  Google Scholar 

  4. Erway, J.B., Gill, P.E.: A subspace minimization method for the trust-region step. SIAM J. Optim. 20(3), 1439–1461 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gardašević Filipović, M.: A trust region method using subgradient for minimizing a nondifferentiable function. Yugosl. J. Oper. Res. 19(2), 249–262 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  10. Xi, H., Shi, B.C.: A trust region method for multiobjective programming without constraints. Math. Appl. (Wuhan) 13(3), 67–69 (2000)

    MATH  MathSciNet  Google Scholar 

  11. Simon, H.: A behavioral model of rational choice. Q. J. Econ. 69(1), 99–118 (1955)

    Article  Google Scholar 

  12. Soubeyran, A.: Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviors. Mimeo (2009)

  13. Soubeyran, A.: Variational rationality and the unsatisfied man: routines and the course pursuit between aspirations, capabilities and beliefs. Mimeo (2010)

  14. Attouch, H., Soubeyran, A.: Inertia and reactivity in decision making as cognitive variational inequalities. J. Convex Anal. 13(2), 207–224 (2006)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  22. Moreno, F., Oliveira, P., Sou Beyran, A.: Proximal algorithm with quasi distance. Application to habits formation. Optimization (2011)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  24. Godal, O., Flam, S., Soubeyran, A.: Gradient differences and bilateral barters. Optimization (2012)

  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)

    MATH  MathSciNet  Google Scholar 

  26. Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51(3), 479–494 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  28. Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (1999)

    Book  MATH  Google Scholar 

  29. Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1995)

    MATH  Google Scholar 

  30. Shi, Z.J., Guo, J.H.: A new trust region method for unconstrained optimization. J. Comput. Appl. Math. 213(2), 509–520 (2008)

    Article  MATH  MathSciNet  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 

  32. Bandura, A., Schunk, D.: Cultivating competence, self efficacy, and intrinsic interest through proximal self motivation. J. Pers. Soc. Psychol. 41, 586–598 (1981)

    Article  Google Scholar 

  33. Brisoux, J.: Le phénomène des ensembles évoqués: une étude empirique des dimensions contenu et taille. Thèse, Université Laval (1995)

  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)

  36. Oliver, R.: Satisfaction: A Behavioral Perspective on the Consumer. McGraw-Hill, New York (2011)

    Google Scholar 

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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.

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Authors and Affiliations

  1. DCC-CI, Federal University of Paraíba, Campus I., CEP: 58.051-900, João Pessoa-Paraíba, Brazil

    Kely D. V. Villacorta

  2. PESC-COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Paulo R. Oliveira

  3. Centro de Tecnologia, Cidade Universitária, CEP: 21.941-97, Bloco H, Sala 319, 68511, Rio de Janeiro, Brazil

    Paulo R. Oliveira

  4. Aix-Marseille University (Aix-Marseille School of Economics), CNRS & EHESS, Chateau Lafarge, 13290, Les Milles, France

    Antoine Soubeyran

Authors
  1. Kely D. V. Villacorta
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  2. Paulo R. Oliveira
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  3. Antoine Soubeyran
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Correspondence to Kely D. V. Villacorta.

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Communicated by Johannes Jahn.

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Villacorta, K.D.V., Oliveira, P.R. & Soubeyran, A. A Trust-Region Method for Unconstrained Multiobjective Problems with Applications in Satisficing Processes. J Optim Theory Appl 160, 865–889 (2014). https://doi.org/10.1007/s10957-013-0392-7

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  • DOI: https://doi.org/10.1007/s10957-013-0392-7

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