×

LP well-posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. (English) Zbl 1474.90461

Summary: The purpose of this paper is introduce several types of Levitin-Polyak well-posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. Base on criterion and characterizations for these types of Levitin-Polyak well-posedness we argue on diameters and Kuratowski’s, Hausdorff’s, or Istrǎtescus measures of noncompactness of approximate solution sets under suitable conditions, and we prove the Levitin-Polyak well-posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. Obtain a gap function for bilevel vector equilibrium problems with equilibrium constraints using the nonlinear scalarization function and consider relations between these types of LP well-posedness for bilevel vector optimization problems with equilibrium constraints and these types of Levitin-Polyak well-posedness for bilevel vector equilibrium problems with equilibrium constraints under suitable conditions; we prove the Levitin-Polyak well-posedness for bilevel equilibrium and optimization problems with equilibrium constraints.

MSC:

90C31 Sensitivity, stability, parametric optimization
49K40 Sensitivity, stability, well-posedness
90C29 Multi-objective and goal programming
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C48 Programming in abstract spaces
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Tikhonov, A. N., On the stability of the functional optimization problem, USSR Computational Mathematics and Mathematical Physics, 6, 4, 28-33, (1966) · Zbl 0212.23803
[2] Levitin, E. S.; Polyak, B. T., Convergence of minimizing sequences in conditional extremum problems, Soviet Mathematics Doklady, 7, 764-767, (1966) · Zbl 0161.07002
[3] Fang, Y.-P.; Hu, R., Parametric well-posedness for variational inequalities defined by bifunctions, Computers & Mathematics with Applications, 53, 8, 1306-1316, (2007) · Zbl 1168.49307
[4] Lignola, M. B.; Morgan, J., Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution, Journal of Global Optimization, 16, 1, 57-67, (2000) · Zbl 0960.90079
[5] Lignola, M. B., Well-posedness and \(L\)-well-posedness for quasivariational inequalities, Journal of Optimization Theory and Applications, 128, 1, 119-138, (2006) · Zbl 1093.49005
[6] Lucchetti, R., Recent Developments in Well-Posed Variational Problems, (1995), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 0823.00006
[7] Lucchetti, R.; Patrone, F., A characterization of Tyhonov well-posedness for minimum problems, with applications to variational inequalities, Numerical Functional Analysis and Optimization, 3, 4, 461-476, (1981) · Zbl 0479.49025
[8] Lucchetti, R.; Patrone, F., Hadamard and Tyhonov well-posedness of a certain class of convex functions, Journal of Mathematical Analysis and Applications, 88, 1, 204-215, (1982) · Zbl 0487.49013
[9] Margiocco, M.; Patrone, F.; Pusillo Chicco, L., A new approach to Tikhonov well-posedness for Nash equilibria, Optimization, 40, 4, 385-400, (1997) · Zbl 0881.90136
[10] Margiocco, M.; Patrone, F.; Pusillo Chicco, L., Metric characterizations of Tikhonov well-posedness in value, Journal of Optimization Theory and Applications, 100, 2, 377-387, (1999) · Zbl 0915.90271
[11] Margiocco, M.; Patrone, F.; Pusillo, L., On the Tikhonov well-posedness of concave games and Cournot oligopoly games, Journal of Optimization Theory and Applications, 112, 2, 361-379, (2002) · Zbl 1011.91004
[12] Morgan, J., Approximations and well-posedness in multicriteria games, Annals of Operations Research, 137, 257-268, (2005) · Zbl 1138.91407
[13] Konsulova, A. S.; Revalski, J. P., Constrained convex optimization problems—well-posedness and stability, Numerical Functional Analysis and Optimization, 15, 7-8, 889-907, (1994) · Zbl 0830.90119
[14] Beer, G.; Lucchetti, R., The epi-distance topology: continuity and stability results with applications to convex optimization problems, Mathematics of Operations Research, 17, 3, 715-726, (1992) · Zbl 0767.49011
[15] Huang, X. X.; Yang, X. Q., Levitin-Polyak well-posedness of constrained vector optimization problems, Journal of Global Optimization, 37, 2, 287-304, (2007) · Zbl 1149.90133
[16] Li, S. J.; Li, M. H., Levitin-Polyak well-posedness of vector equilibrium problems, Mathematical Methods of Operations Research, 69, 1, 125-140, (2009) · Zbl 1190.90266
[17] Huang, N.-J.; Long, X.-J.; Zhao, C.-W., Well-posedness for vector quasi-equilibrium problems with applications, Journal of Industrial and Management Optimization, 5, 2, 341-349, (2009) · Zbl 1192.49028
[18] Li, M. H.; Li, S. J.; Zhang, W. Y., Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems, Journal of Industrial and Management Optimization, 5, 4, 683-696, (2009) · Zbl 1191.90079
[19] Outrata, J. V., A generalized mathematical program with equilibrium constraints, SIAM Journal on Control and Optimization, 38, 5, 1623-1638, (2000) · Zbl 0968.49012
[20] Mordukhovich, B. S., Characterizations of linear suboptimality for mathematical programs with equilibrium constraints, Mathematical Programming, 120, 1, 261-283, (2009) · Zbl 1169.90021
[21] Lignola, M. B.; Morgan, J., \(\alpha\)-well-posedness for Nash equilibria and for optimization problems with Nash equilibrium constraints, Journal of Global Optimization, 36, 3, 439-459, (2006) · Zbl 1105.49029
[22] Bao, T. Q.; Gupta, P.; Mordukhovich, B. S., Necessary conditions in multiobjective optimization with equilibrium constraints, Journal of Optimization Theory and Applications, 135, 2, 179-203, (2007) · Zbl 1146.90508
[23] Anh, L. Q.; Khanh, P. Q.; Van, D. T. M., Well-posedness under relaxed semicontinuity for bilevel equilibrium and optimization problems with equilibrium constraints, Journal of Optimization Theory and Applications, 153, 1, 42-59, (2012) · Zbl 1254.90244
[24] Ansari, Q. H.; Giannessi, F., Vector equilibrium problems and vector variational inequalities, Vector Variational Inequalities and Vector Equilibria. Vector Variational Inequalities and Vector Equilibria, Mathematical Theories, 116, (2000), Dordrecht, The Netherlands: Kluwer, Dordrecht, The Netherlands
[25] Giannessi, F., Vector Variational Inequalities and Vector Equilibria. Vector Variational Inequalities and Vector Equilibria, Nonconvex Optimization and its Applications, 38, (2000), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 0952.00009
[26] Chen, G. Y.; Yang, X. Q.; Yu, H., A nonlinear scalarization function and generalized quasi-vector equilibrium problems, Journal of Global Optimization, 32, 4, 451-466, (2005) · Zbl 1130.90413
[27] Li, S. J.; Teo, K. L.; Yang, X. Q., Generalized vector quasi-equilibrium problems, Mathematical Methods of Operations Research, 61, 3, 385-397, (2005) · Zbl 1114.90114
[28] Li, S. J.; Teo, K. L.; Yang, X. Q.; Wu, S. Y., Gap functions and existence of solutions to generalized vector quasi-equilibrium problems, Journal of Global Optimization, 34, 3, 427-440, (2006) · Zbl 1090.49014
[29] Ansari, Q. H.; Flores-Bazán, F., Generalized vector quasi-equilibrium problems with applications, Journal of Mathematical Analysis and Applications, 277, 1, 246-256, (2003) · Zbl 1022.90023
[30] Bianchi, M.; Hadjisavvas, N.; Schaible, S., Vector equilibrium problems with generalized monotone bifunctions, Journal of Optimization Theory and Applications, 92, 3, 527-542, (1997) · Zbl 0878.49007
[31] Song, W., On generalized vector equilibrium problems, Applied Mathematics Letters, 12, 53-56, (2002)
[32] Kuratowski, C., Topology. Panstwowe Wydawnicto Naukowa, 1, (1958), Warszawa, Poland
[33] Daneš, J., On the Istrătescu measure of noncompactness, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, 16, 4, 403-406, (1972) · Zbl 0293.54038
[34] Banaś, J.; Goebel, K., Measures of Noncompactness in Banach Spaces. Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, 60, (1980), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0441.47056
[35] Rakočević, V., Measures of noncompactness and some applications, Filomat, 12, 87-120, (1998) · Zbl 1009.47047
[36] Chen, G.-Y.; Huang, X.; Yang, X., Vector Optimization, Set-Valued and Variational Analysis. Vector Optimization, Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems, 541, (2005), Berlin, Germany: Springer, Berlin, Germany · Zbl 1104.90044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.