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Well-Posedness Under Relaxed Semicontinuity for Bilevel Equilibrium and Optimization Problems with Equilibrium Constraints

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Abstract

Bilevel equilibrium and optimization problems with equilibrium constraints are considered. We propose a relaxed level closedness and use it together with pseudocontinuity assumptions to establish sufficient conditions for well-posedness and unique well-posedness. These conditions are new even for problems in one-dimensional spaces, but we try to prove them in general settings. For problems in topological spaces, we use convergence analysis while for problems in metric cases we argue on diameters and Kuratowski’s, Hausdorff’s, or Istrǎtescu’s measures of noncompactness of approximate solution sets. Besides some new results, we also improve or generalize several recent ones in the literature. Numerous examples are provided to explain that all the assumptions we impose are very relaxed and cannot be dropped.

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References

  1. Hadamard, J.: Sur le problèmes aux dérivees partielles et leur signification physique. Bull. Univ. Princeton 13, 49–52 (1902)

    MathSciNet  Google Scholar 

  2. Tikhonov, A.N.: On the stability of the functional optimization problem. USSR Comput. Math. Math. Phys. 6, 28–33 (1966)

    Article  Google Scholar 

  3. Revalski, J.P.: Hadamard and strong well-posedness for convex programs. SIAM J. Optim. 7, 519–526 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ioffe, A., Lucchetti, R.E.: Typical convex program is very well-posed. Math. Program., Ser. B 104, 483–499 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ioffe, A., Lucchetti, R.E., Revalski, J.P.: Almost every convex or quadratic programming problem is well-posed. Math. Oper. Res. 29, 369–382 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Crespi, G.P., Guerraggio, A., Rocca, M.: Well-posedness in vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 132, 213–226 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Morgan, J., Scalzo, V.: Discontinuous but well-posed optimization problems. SIAM J. Optim. 17, 861–870 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zolezzi, T.: Well-posedness criteria in optimization with applications to the calculus of variations. Nonlinear Anal. 25, 437–453 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zolezzi, T.: Well-posedness and optimization under perturbations. Ann. Oper. Res. 101, 351–361 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zolezzi, T.: On well-posedness and conditioning in optimization. Z. Angew. Math. Mech. 84, 435–443 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Morgan, J., Scalzo, V.: Pseudocontinuity in optimization and nonzero sum games. J. Optim. Theory Appl. 120, 181–197 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fang, Y.P., Hu, R., Huang, N.J.: Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints. Comput. Math. Appl. 55, 89–100 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ceng, L.C., Hadjisavvas, N., Schaible, S., Yao, J.C.: Well-posedness for mixed quasivariational-like inequalities. J. Optim. Theory Appl. 139, 109–225 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fang, Y.P., Huang, N.J., Yao, J.C.: Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems. J. Glob. Optim. 41, 117–133 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lignola, M.B.: Well-posedness and L-well-posedness for quasivariational inequalities. J. Optim. Theory Appl. 128, 119–138 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yu, J., Yang, H., Yu, C.: Well-posed Ky Fan’s point, quasivariational inequality and Nash equilibrium points. Nonlinear Anal. 66, 777–790 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lignola, M.B., Morgan, J.: α-Well-posedness for Nash equilibria and for optimization problems with Nash equilibrium constraints. J. Glob. Optim. 36, 439–459 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Margiocco, M., Patrone, F., Pusillo Chicco, L.: A new approach to Tikhonov well-posedness for Nash equilibria. Optimization 40, 385–400 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Anh, L.Q., Khanh, P.Q., Van, D.T.M., Yao, J.C.: Well-posedness for vector quasiequilibria. Taiwan. J. Math. 13, 713–737 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Kimura, K., Liou, Y.C., Wu, S.Y., Yao, J.C.: Well-posedness for parametric vector equilibrium problems with applications. J. Ind. Manag. Optim. 4, 313–327 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Outrata, J.V.: A generalized mathematical program with equilibrium constraints. SIAM J. Control Optim. 38, 1623–1638 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mordukhovich, B.S.: Characterizations of linear suboptimality for mathematical programs with equilibrium constraints. Math. Program., Ser. B 120, 261–283 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lignola, M.B., Morgan, J.: Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution. J. Glob. Optim. 16, 57–67 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bao, T.Q., Gupta, P., Mordukhovich, B.S.: Necessary conditions in multiobjective optimization with equilibrium constraints. J. Optim. Theory Appl. 135, 179–203 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Anh, L.Q., Khanh, P.Q.: Semicontinuity of the solution sets to parametric quasiequilibrium problems. J. Math. Anal. Appl. 294, 699–711 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Khanh, P.Q., Luu, L.M.: Lower semicontinuity and upper semicontinuity of the solution sets to parametric multivalued quasivariational inequalities. J. Optim. Theory Appl. 133, 329–339 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Anh, L.Q., Khanh, P.Q.: Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks, II: Lower semicontinuities, applications. Set-Valued Anal. 16, 943–960 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hai, N.X., Khanh, P.Q.: Existence of solutions to general quasiequilibrium problems and applications. J. Optim. Theory Appl. 133, 317–327 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hai, N.X., Khanh, P.Q.: The solution existence of general variational inclusion problems. J. Math. Anal. Appl. 328, 1268–1277 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hai, N.X., Khanh, P.Q., Quan, N.H.: On the existence of solutions to quasivariational inclusion problems. J. Glob. Optim. 45, 565–581 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Birbil, S.I., Bouza, G., Frenk, J.B.G., Still, G.: Equilibrium constrained optimization problems. Eur. J. Oper. Res. 169, 1108–1128 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mordukhovich, B.S.: Equilibrium problems with equilibrium constraints via multiobjective optimization. Optim. Methods Softw. 19, 479–492 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mordukhovich, B.S.: Optimization and equilibrium problems with equilibrium constraints in infinite-dimensional spaces. Optimization 57, 1–27 (2008)

    Article  MathSciNet  Google Scholar 

  34. Mordukhovich, B.S.: Multiobjective optimization problems with equilibrium constraints. Math. Program. 117, 331–354 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Outrata, J.V., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic, Dordrecht (1998)

    MATH  Google Scholar 

  36. Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Nonlinear Parametric Optimization. Birkhäuser, Basel (1983)

    Google Scholar 

  37. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, 3rd edn. Springer, Berlin (2009)

    MATH  Google Scholar 

  38. Anh, L.Q., Khanh, P.Q., Van, D.T.M.: Well-posedness without semicontinuity for parametric quasiequilibria and quasioptimization. Comput. Math. Appl. 62, 2045–2057 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Khanh, P.Q., Luc, D.T., Tuan, N.D.: Local uniqueness of solutions for equilibrium problems. Adv. Nonlinear Var. Inequal. 9, 13–27 (2006)

    MathSciNet  MATH  Google Scholar 

  40. Khanh, P.Q., Tung, L.T.: Local uniqueness of solutions to vector equilibrium problems using approximations. J. Optim. Theory Appl. Submitted for publication

  41. Danes, J.: On the Istrǎtescu measure of noncompactness. Bull. Math. Soc. R.S. Roumanie 16, 403–406 (1972)

    MathSciNet  Google Scholar 

  42. Banas, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Marcel Dekker, New York-Basel (1980)

    MATH  Google Scholar 

  43. Rakocěvíc, V.: Measures of noncompactness and some applications. Filomat 12, 87–120 (1998)

    MATH  Google Scholar 

  44. Jongen, H.T., Weber, G.W.: Nonlinear optimization: characterization of structural stability. J. Glob. Optim. 1, 47–64 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Teacher College, Cantho University, Cantho, Vietnam

    L. Q. Anh

  2. Department of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Vietnam

    P. Q. Khanh

  3. Department of Mathematics, Cantho College, Cantho, Vietnam

    D. T. M. Van

Authors
  1. L. Q. Anh
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  2. P. Q. Khanh
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  3. D. T. M. Van
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Corresponding author

Correspondence to P. Q. Khanh.

Additional information

Communicated by Aram Arutyunov.

This work was supported by National Foundation for Science and Technology Development of Vietnam. The authors would like to thank the anonymous referees for their valuable remarks and suggestions, which have helped to improve the paper.

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Anh, L.Q., Khanh, P.Q. & Van, D.T.M. Well-Posedness Under Relaxed Semicontinuity for Bilevel Equilibrium and Optimization Problems with Equilibrium Constraints. J Optim Theory Appl 153, 42–59 (2012). https://doi.org/10.1007/s10957-011-9963-7

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