Taiwanese Journal of Mathematics

WELL-POSEDNESS FOR VECTOR QUASIEQUILIBRIA

Lam Quoc Anh, Phan Quoc Khanh, Dang Thi My Van, and Jen-Chih Yao

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Abstract

We consider well-posedness under perturbations of vector quasiequilibrium and bilevel-equilibrium problems. This kind of well-posedness relates Hadamard and Tikhonov well-posedness notions to sensitivity analysis and we apply largely techniques of the latter to establish sufficient conditions for wellposedness under perturbations. We also propose several new semicontinuity and quasiconvexity notions to weaken the imposed assumptions. Our results are new or include as special cases recent existing results. Many examples are provided for the illustration purpose.

Article information

Source
Taiwanese J. Math., Volume 13, Number 2B (2009), 713-737.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405398

Digital Object Identifier
doi:10.11650/twjm/1500405398

Mathematical Reviews number (MathSciNet)
MR2510823

Zentralblatt MATH identifier
1176.49030

Subjects
Primary: 49K40: Sensitivity, stability, well-posedness [See also 90C31] 90C31: Sensitivity, stability, parametric optimization 65K99: None of the above, but in this section

Keywords
well-posedness unique well-posedness perturbations vector quasiequilibrium problems bilevel problems cone-quasi semicontinuity cone-level convexity

Citation

Anh, Lam Quoc; Khanh, Phan Quoc; Van, Dang Thi My; Yao, Jen-Chih. WELL-POSEDNESS FOR VECTOR QUASIEQUILIBRIA. Taiwanese J. Math. 13 (2009), no. 2B, 713--737. doi:10.11650/twjm/1500405398. https://projecteuclid.org/euclid.twjm/1500405398


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References

  • L. Q. Anh and P. Q. Khanh, Semicontinuity of the solution sets of parametric multivalued vector quasiequilibrium problems, J. Math. Anal. Appl., 294 (2004), 699-711.
  • L. Q. Anh and P. Q. Khanh, On the stability of the solution sets of general multivalued vector quasiequilibrium problems, J. Optim. Theory Appl., 135 (2007), 271-284.
  • L. Q. Anh and P. Q. Khanh, Various kinds of semicontinuity and the solutions ets of parametric multivalued symmetric vector quasiequilibrium problems, J. Glob. Optim., 41 (2008), 539-558.
  • L. Q. Anh and P. Q. Khanh, Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems, Numer. Funct. Anal. Optim., 29 (2008), 24-42.
  • H. Attouch and R. Wets, Quantitative stability of variational systems, Trans. Amer. Math. Soc., 328 (1991), 695-730.
  • G. Beer and R. Lucchetti, The epi-distance topology, Math. Oper. Res., 17 (1992), 715-726.
  • L. C. Ceng, N. Hadjisavvas, S. Schaible and J. C. Yao, Well-posedness for mixed quasivariational-like inequalities, J. Optim. Theory Appl., 139 (2008), 109-125.
  • L. C. Ceng and J. C. Yao, Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed point problems, Nonlinear Anal. TMA, 69 (2008), 4585-4603.
  • G. P. Crespi, A. Guerraggio and M. Rocca, Well-posedness in vector optimization problems and vector variational inequalities, J. Optim. Theory Appl., 132 (2007), 213-226.
  • J. Demmel, On condition numbers and the distance of the nearest ill-posed problem, Numer. Math., 51 (1987), 251-289.
  • Y. P. Fang, N. J. Huang and J. C. Yao, Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems, J. Glob. Optim., 41 (2008), 117-133.
  • Y. P. Fang, R. Hu and N.J. Huang, Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints, Computers Math. Appl., 55 (2008), 89-100.
  • M. Furi and A. Vignoli, About well-posed optimization problem for functionals in metric spaces, J. Optim. Theory Appl., 5 (1970), 225-229.
  • A. Göpfert, C. Tammer, H. Riahi and C. Z\v alinescu, Variational Methods in Partially Ordered Spaces, Springer, Berlin, Germany, 2003.
  • J. Hadamard, Sur le problèmes aux dérivees partielles et leur signification physique, Bull. Univ. Princeton, 13 (1902), 49-52.
  • X. X. Huang and X. Q. Yang, Gereralized Levitin-Polyak well-posedness in constrained optimization, SIAM J. Optim., 17 (2006), 243-258.
  • X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness of constrained vector optimization problems, J. Glob. Optim., 37 (2007), 287-304.
  • A. Ioffe and R. E. Lucchetti, Typical convex program is very well-posed, Math. Prog., Ser. B, 104 (2005), 483-499.
  • A. Ioffe, R. E. Lucchetti and J. P. Revalski, A variational principle for problems with functional constraints, SIAM J. Optim., 12 (2001), 461-478.
  • A. Ioffe, R. E. Lucchetti and J. P. Revalski, Almost every convex or quadratic programming problem is well-posed, Math. Oper. Res., 29 (2004), 369-382.
  • P. S. Kenderov and J. P. Revalski, Generic well-posedness of optimization problems and Banach-Mazur game, in: Recent Developments in Well-Posed Variational Problems, R. Lucchetti and J. P. Revalski eds, Math. Appl., 331, Kluwer Academic, Dordrecht, Netherlands, 1995, 117-136.
  • K. Kimura, Y. C. Liou, S. Y. Wu and J. C. Yao, Well-posedness for parametric vector equilibbrium problems with applications, J. Indust. Manag. Optim., to appear.
  • A. S. Konsulova and J. P. Revalski, Constrained convex optimization problems - well-posedness and stability, Numer. Funct. Anal. Optim., 15 (1994), 889-907.
  • B. Lemaire, C. Ould Ahmed Salem and J. P. Revalski, Well-posedness of variational problems with applications to staircase methods, C. R. Acad. Sci. Paris, 332, ser. 1 (2001), 943-948.
  • B. Lemaire, C. Ould Ahmed Salem and J. P. Revalski, Well-posedness by perturbations of variational problems, J. Optim. Theory Appl., 115 (2002), 345-368.
  • E. S. Levitin and B. T. Polyak, Convergence of minimizing sequences in conditional extremum problems, Soviet Math. Dokl. 7 (1966), 764-767.
  • M. B. Lignola, Well-posedness and $L$-well-posedness for quasivariational inequalities, J. Optim. Theory Appl., 128 (2006), 119-138.
  • M. B. Lignola and J. Morgan, Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution, J. Glob. Optim., 16 (2000), 57-67.
  • M. B. Lignola and J. Morgan, $\alpha$-Well-posedness for Nash equilibria and for optimization problems with Nash equilibrium constraints, J. Glob. Optim., 36 (2006), 439-459.
  • R. Lucchetti and F. Patrone, A characterization of Tikhonov well-posedness for minimum problems, with applications to variational inequalities, Numer. Funct. Anal. Optim., 3 (1981), 461-476.
  • R. Lucchetti and F. Patrone, Hadamard and Tikhonov well-posedness of a certain class of convex functions, J. Math. Anal. Appl., 88 (1982), 204-215.
  • M. Margiocco, F. Patrone and L. Pusillo Chicco, A new approach to Tikhonov well-posedness for Nash equilibria, Optim., 40 (1997), 385-400.
  • J. Morgan and V. Scalzo, Discontinuous but well-posed optimization problems, SIAM J. Optim., 17 (2006), 861-870.
  • J. P. Revalski, Hadamard and strong well-posedness for convex programs, SIAM J. Optim., 7 (1997), 519-526.
  • J. P. Revalski, Gereric properties concerning well-posed optimization problems, C. R. Acad. Bulgaria Sci., 38 (1985), 1431-1434.
  • A. N. Tikhonov, On the stability of the functional optimization problem, Soviet Comput. Math. Math. Phys., 6 (1966), 28-33.
  • T. Zolezzi, Well-posedness criteria in optimization with applications to the calculus of variations, Nonlinear Anal. TMA, 25 (1995), 437-453.
  • T. Zolezzi, Well-posedness and optimization under perturbations, Annals Oper. Res., 101 (2001), 351-361.
  • T. Zolezzi, On the distance theorem in quadratic optimization, J. Convex Anal., 9 (2002), 693-700.
  • T. Zolezzi, Condition number theorems in optimization, SIAM J. Optim., 14 (2003), 507-516.
  • T. Zolezzi, On well-posedness and conditioning in optimization, ZAMM Z. Angew. Math. Mech., 84 (2004), 435-443.