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Journal of Global Optimization

, Volume 56, Issue 4, pp 1501–1513 | Cite as

Convergence of a class of penalty methods for constrained scalar set-valued optimization

  • X. X. Huang
Article

Abstract

In this paper, we study a class of penalty methods for a class of constrained scalar set-valued optimization problems. We establish an equivalence relation between the lower semicontinuity at the origin of the optimal value function of the perturbed problem and the convergence of the penalty methods. Some sufficient conditions that guarantee the convergence of the penalty methods are also derived.

Keywords

Constrained scalar set-valued optimization Penalty methods Convergence Lower semicontinuity of a function Upper semicontinuity of a set-valued map 

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References

  1. 1.
    Bertsekas D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)Google Scholar
  2. 2.
    Burke J.V.: Calmness and exact penalization. SIAM J. Control Optim. 29, 493–497 (1991)CrossRefGoogle Scholar
  3. 3.
    Chen G.Y., Huang X.X., Yang X.Q.: Vector Optimization, Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)Google Scholar
  4. 4.
    Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)Google Scholar
  5. 5.
    Fiacco A., McCormic G.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York (1968)Google Scholar
  6. 6.
    Fiacco A., Ishizuka V.: Sensitivity and stability analysis for nonlinear programming. Ann. Oper. Res. 27, 215–235 (1990)CrossRefGoogle Scholar
  7. 7.
    Fletcher R.: Practical Methods of Optimization. Wiley, New York (1987)Google Scholar
  8. 8.
    Ha, T.X.D.: Optimality conditions for several types of efficient solutions of set-valued optimization problems. In: Pardalos, P., Rassias, Th.M., Khan, A.A. (eds.) Nonlinear Analysis and Variational Problems, pp. 305–324. Springer, Heidelberg (2010)Google Scholar
  9. 9.
    Hamel A.H., Heyde F.: Duality for set-valued measures of risk. SIAM J. Financ. Math. 1, 66–95 (2010)CrossRefGoogle Scholar
  10. 10.
    Huang, X.X.: Calmness and exact penalization in constrained scalar set-valued optimization. J. Optim. Theory Appl. doi: 10.1007/s10957-012-9998-4 (2012)
  11. 11.
    Huang X.X., Yang X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28, 533–552 (2003)CrossRefGoogle Scholar
  12. 12.
    Huang X.X., Yang X.Q.: Characterizations of the nonemptiness and compactness of the set of weakly efficient solutions for convex vector optimization and applications. J. Math. Anal. Appl. 264, 270–287 (2001)CrossRefGoogle Scholar
  13. 13.
    Huang X.X., Yang X.Q., Teo K.L.: A lower order penalization approach to nonlinear semidefinite programs. J. Optim. Theory Appl. 132, 1–20 (2007)CrossRefGoogle Scholar
  14. 14.
    Huang X.X., Yang X.Q., Zhu D.L.: A sequential smooth penalization approach to mathematical programs with complementarity constraints. Numer. Funct. Anal. Optim. 27, 71–98 (2006)CrossRefGoogle Scholar
  15. 15.
    Huang, X.X., Yao, J.C.: Characterizations for the nonemptiness and compactness of the solution sets of convex set-valued optimization problems. J. Glob. Optim. doi: 10.1007/s10898-012-9846-y (2012)
  16. 16.
    Jahn J.: Vector Optimization. Theory, Applications, and Extensions. Springer, Berlin (2004)CrossRefGoogle Scholar
  17. 17.
    Luc D.T.: Theory of Vector Optimization. Springer, Berlin (1989)CrossRefGoogle Scholar
  18. 18.
    Luo Z.Q., Pang J.S., Ralph D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, New York (1996)CrossRefGoogle Scholar
  19. 19.
    Rosenberg E.: Exact penalty functions and stability in locally Lipschitz programming. Math. Program. 30, 340–356 (1984)CrossRefGoogle Scholar
  20. 20.
    Rubinov A.M., Huang X.X., Yang X.Q.: Zero duality gap property and lower semicontinuity of perturbation function. Math. Oper. Res. 27, 775–791 (2002)CrossRefGoogle Scholar
  21. 21.
    Rubinov A.M., Yang X.Q.: Lagrange-Type Functions in Constrained Non-convex Optimization. Kluwer, New York (2003)Google Scholar
  22. 22.
    Sawaragi Y., Nakayama H., Tanino T.: Theory of Multiobjective Optimization. Academic Press, New York (1985)Google Scholar
  23. 23.
    Yang X.Q., Huang X.X.: Lower order penalty methods for mathematical programs with complementarity constraints. Optim. Methods Softw. 19, 693–720 (2004)CrossRefGoogle Scholar
  24. 24.
    Wang C.Y., Yang X.Q., Yang X.M.: Nonlinear Lagrange duality theorems and penalty function methods in continuous optimization. J. Glob. Optim. 27, 473–484 (2003)CrossRefGoogle Scholar
  25. 25.
    Wang C.Y., Yang X.Q., Yang X.M.: Unified nonlinear Lagrangian approach to duality and optimal path. J. Optim. Theory Appl. 135, 85–100 (2007)CrossRefGoogle Scholar
  26. 26.
    Wang C.Y., Li D.: Unified theory of augmented Lagrangian methods for constrained global optimization. J. Glob. Optim. 44, 433–458 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  1. 1.School of Economics and Business AdministrationChongqing UniversityChongqingChina

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