向量优化问题的适定性研究
|
摘要
本文主要研究了集优化问题、对称向量拟平衡问题及广义向量拟平衡问题的适定性,向量值优化问题的Hadamard适定性,以及凸向量值优化问题和凸集值优化问题的适定性和稳定性。全文共分八章,具体内容如下:
在第一章,我们介绍了标量优化问题、向量值优化问题、集值优化问题、变分不等式问题以及变分平衡问题适定性的研究概况,向量值优化稳定性的研究概况以及稳定性和适定性的关系,并且阐述了本文的选题动机和主要工作。
在第二章,介绍本文中频繁使用的一些基本概念。
在第三章,对集优化问题引入了三种类型的适定性。利用广义的Gerstewitz’s函数,分别建立了这三种类型的适定性和对应的标量优化问题之间的等价关系。利用广义压迫函数得出了集优化问题适定性的充要条件。最后,给出了集优化适定性的判别准则。
在第四章,定义了关于对称向量拟平衡问题的Levitin-Polyak适定性。利用集值优化问题不动点问题的适定性得出了对称向量拟平衡问题的Levitin-Polyak适定的充分性条件。
在第五章,在两类广义向量拟平衡问题中引入Levitin-Polyak适定性的概念,然后在该模型中讨论了一些适定性的经典性质,通过两类广义向量拟平衡问题的间隙函数得到了标量优化问题Levitin-Polyak适定性和广义向量拟平衡问题Levitin-Polyak适定性的关系,最后建立了一种集值的Ekeland变分原理,并利用该原理得到了一类广义向量拟平衡问题Levitin-Polyak适定性的充分条件。
在第六章,定义了向量值优化问题的两类Hadamard适定性。利用标量化函数,建立了关于向量值映射序列Gamma-收敛的标量化定理。利用所得的标量化定理,给出了对于向量值优化问题的Hadamard适定性的充分性条件。
在第七章,研究了关于映射序列的P.K.收敛的凸向量值优化问题和凸集值优化问题的稳定性和适定性。讨论了向量映射列的Gamma-收敛,P.K.-收敛,逐点收敛和连续收敛之间的包含关系和等价条件。
在第八章,我们作了一个简要的总结和讨论。
In this thesis, the well-posedness of set optimization problems, symmetric vector quasi-equilibrium problems and generalized vector quasi-equilibrium problems, and Hadamard well-posedness of vector optimization problems are studied. Moreover, the well-posedness and stability properties of convex vector optimization problems and set-valued optimization problems are disscussed, respectively. This thesis is divided into eight chapters. It is organized as follows.
In Chapter 1, the development and researches on the topic of wellposedness of scalar-valued problems, vector-valued problems, set-valued problems, variational inequalities and vector equilibrium problems are described. The study on the stability of vector optimization and the relationship between the stability and well-posedness are discussed. Also, the motivation is given and main works are listed.
In Chapter 2, some definitions, which will be frequently used, are shown.
In Chapter 3, three kinds of well-posedness for set optimization are first introduced. By virtue of a generalized Gerstewitz’s function, the equivalent relations between the three kinds of well-posedness and the well-posedness of three kinds of scalar optimization problems are established, respectively. Then, sufficient and necessary conditions of well-posedness for set optimization problems are obtained by using a generalized forcing function. Finally, various criteria and characterizations of well-posedness are given for set optimization problems.
In Chapter 4, a generalized Levitin-Polyak well-posedness of symmetric vector quasi-equilibrium problems is defined. By verifying the result of Hadamard well-posedness of set-valued fixed point problems, sufficient conditions for the generalized Levitin-Polyak well-posedness of symmetric vector quasi-equilibrium problems are given.
In Chapter 5, the definitions of Levitin-Polyak well-posedness for two classes of generalized vector quasi-equilibrium problems are introduced. Then, some classical criteria and characterizations of the Levitin-Polyak well-posedness are investigated. And by virtue of gap functions for the generalized vector quasi-equilibrium problems, some equivalent relations are obtained between the Levitin-Polyak well-posedness for optimization problems and the Levitin-Polyak well-posedness for the generalized vector quasi-equilibrium problems. Finally, a set-valued version of Ekeland's variational principle is derived and applied to establish a sufficient condition for Levitin-Polyak well-posedness of a class of generalized vector quasi-equilibrium problems.
In Chapter 6, two kinds of Hadamard well-posedness for vector-valued optimization problems are introduced. By virtue of scalarization functions, the scalarization theorems of convergence for sequences of vector-valued functions are established. Sufficient conditions of Hadamard well-posedness for vector optimization problems are obtained by using the scalarization theorems.
In Chapter 7, well-posedness and stability for convex vector-valued optimization problems and set-valued optimization problems are introduced. The relationship among Gamma-convergence, P.K.convergence, pointwise convergence and continuous convergence of sequences of vector-valued functions are investigated.
In Chapter 8, the results of this thesis are summarized and some discussions are made.
在第一章,我们介绍了标量优化问题、向量值优化问题、集值优化问题、变分不等式问题以及变分平衡问题适定性的研究概况,向量值优化稳定性的研究概况以及稳定性和适定性的关系,并且阐述了本文的选题动机和主要工作。
在第二章,介绍本文中频繁使用的一些基本概念。
在第三章,对集优化问题引入了三种类型的适定性。利用广义的Gerstewitz’s函数,分别建立了这三种类型的适定性和对应的标量优化问题之间的等价关系。利用广义压迫函数得出了集优化问题适定性的充要条件。最后,给出了集优化适定性的判别准则。
在第四章,定义了关于对称向量拟平衡问题的Levitin-Polyak适定性。利用集值优化问题不动点问题的适定性得出了对称向量拟平衡问题的Levitin-Polyak适定的充分性条件。
在第五章,在两类广义向量拟平衡问题中引入Levitin-Polyak适定性的概念,然后在该模型中讨论了一些适定性的经典性质,通过两类广义向量拟平衡问题的间隙函数得到了标量优化问题Levitin-Polyak适定性和广义向量拟平衡问题Levitin-Polyak适定性的关系,最后建立了一种集值的Ekeland变分原理,并利用该原理得到了一类广义向量拟平衡问题Levitin-Polyak适定性的充分条件。
在第六章,定义了向量值优化问题的两类Hadamard适定性。利用标量化函数,建立了关于向量值映射序列Gamma-收敛的标量化定理。利用所得的标量化定理,给出了对于向量值优化问题的Hadamard适定性的充分性条件。
在第七章,研究了关于映射序列的P.K.收敛的凸向量值优化问题和凸集值优化问题的稳定性和适定性。讨论了向量映射列的Gamma-收敛,P.K.-收敛,逐点收敛和连续收敛之间的包含关系和等价条件。
在第八章,我们作了一个简要的总结和讨论。
In this thesis, the well-posedness of set optimization problems, symmetric vector quasi-equilibrium problems and generalized vector quasi-equilibrium problems, and Hadamard well-posedness of vector optimization problems are studied. Moreover, the well-posedness and stability properties of convex vector optimization problems and set-valued optimization problems are disscussed, respectively. This thesis is divided into eight chapters. It is organized as follows.
In Chapter 1, the development and researches on the topic of wellposedness of scalar-valued problems, vector-valued problems, set-valued problems, variational inequalities and vector equilibrium problems are described. The study on the stability of vector optimization and the relationship between the stability and well-posedness are discussed. Also, the motivation is given and main works are listed.
In Chapter 2, some definitions, which will be frequently used, are shown.
In Chapter 3, three kinds of well-posedness for set optimization are first introduced. By virtue of a generalized Gerstewitz’s function, the equivalent relations between the three kinds of well-posedness and the well-posedness of three kinds of scalar optimization problems are established, respectively. Then, sufficient and necessary conditions of well-posedness for set optimization problems are obtained by using a generalized forcing function. Finally, various criteria and characterizations of well-posedness are given for set optimization problems.
In Chapter 4, a generalized Levitin-Polyak well-posedness of symmetric vector quasi-equilibrium problems is defined. By verifying the result of Hadamard well-posedness of set-valued fixed point problems, sufficient conditions for the generalized Levitin-Polyak well-posedness of symmetric vector quasi-equilibrium problems are given.
In Chapter 5, the definitions of Levitin-Polyak well-posedness for two classes of generalized vector quasi-equilibrium problems are introduced. Then, some classical criteria and characterizations of the Levitin-Polyak well-posedness are investigated. And by virtue of gap functions for the generalized vector quasi-equilibrium problems, some equivalent relations are obtained between the Levitin-Polyak well-posedness for optimization problems and the Levitin-Polyak well-posedness for the generalized vector quasi-equilibrium problems. Finally, a set-valued version of Ekeland's variational principle is derived and applied to establish a sufficient condition for Levitin-Polyak well-posedness of a class of generalized vector quasi-equilibrium problems.
In Chapter 6, two kinds of Hadamard well-posedness for vector-valued optimization problems are introduced. By virtue of scalarization functions, the scalarization theorems of convergence for sequences of vector-valued functions are established. Sufficient conditions of Hadamard well-posedness for vector optimization problems are obtained by using the scalarization theorems.
In Chapter 7, well-posedness and stability for convex vector-valued optimization problems and set-valued optimization problems are introduced. The relationship among Gamma-convergence, P.K.convergence, pointwise convergence and continuous convergence of sequences of vector-valued functions are investigated.
In Chapter 8, the results of this thesis are summarized and some discussions are made.
引文
[1] A. N. Tykhonov, On the stability of the functional optimization problem[J]. USSR Computational Mathematics and Mathematical Physics, 1966, 6:28-33.
[2] T. Zolezzi, Extended well-posedness of optimization problems[J]. Journal of Optimization Theory and Applications, 1996, 96:257-266.
[3] A. L. Dontchev, T. Zolezzi, Well-posed Optimization Problems[M]. In: Lecture Notes in Mathematics, 1543, Springer-Verlag, Berlin, 1993.
[4] E. S. Levitin, and B. T. Polyak, Convergence of minimizing sequences in conditional extremum problems[J]. Soviet Mathematics Doklady, 1966, 7:764-767.
[5] X. X. Huang, and X. Q. Yang, Generalized Levitin-Polyak well-posedness in constrained optimization[J]. SIAM Journal on Optimization, 2006, 17(1):243-258.
[6] E. Miglierina, and E. Molho, Well-posedness and stability for abstract spline problems[J]. Journal of Mathematical Analysis and Applications, 2007(333):1058–1069.
[7] T. Zolezzi, Well-posedness and optimization under perturbations[J]. Annals of Operations Research, 2001, 101:351-361.
[8] D. Dentcheva, S. Helbig, On variational principles, level sets, well-posedness andε-solutions in vector optimization[J]. Journal of Optimization Theory and Application, 1996, 89:329-349.
[9] X. X. Huang, Extended well-posedness Properties of vector optimization problems[J]. Journal of Optimization Theory and Applications, 2000, 106(1):165 -182.
[10] X. X. Huang, Extended and strongly extended well-posedness of set-valued optimization problems[J]. Mathematical Methods of Operations Research, 2001, 53:101-116.
[11] X. X. Huang, Pointwise well-posedness of perturbed vector optimization problems in a vector-valued variational principle[J]. Journal of Optimization Theory and Applications, 2001, 108(3):671-686.
[12] E. M. Bednarczuk, An approach to well-posedness in vector optimization problems: consequences to stability[J]. Control and Cybernetics, 1994, 23:107-122.
[13] E. Miglierina and E. Molho, Well-posedness and convexity in vector optimization[J]. Mathematical Methods Operations Reseach, 2003, 58:375-385.
[14] E. Miglierina, E. Molho and M. Rocca, Well-posedness and scalarization in vector optimization[J]. Journal of Optimization Theory and Applications, 2005, 126(2):391-409.
[15] R. Lucchetti, Well-posedness, towards vector optimization[J]. Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, 1987, 294:194-207.
[16] E. Bednarczuk, Well-posedness of vector optimization problems[J]. In: Jahn, J. and Krabs, W. (eds), Recent advances and historical development of vector optimization problems, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 1987, 294:51–61.
[17] P. Loridan, Well-posedness in vector optimization[A]. In: Lucchetti, R. and Revalski, J. (eds), Recent developments in variational well-posedness problems. Mathematics and its Applications, Kluwer Academic Publisher, Dordrecht, 1995, 331:171–192.
[18] Y. P. Fang and N. J. Huang, Increasing-along-rays property, vector optimization and well-posedness[J]. Mathematical Methods of Operations Research, 2007, 65: 99–114.
[19] M. Durea, Scalarization for pointwise well-posed vectorial problems[J]. Mathematical Metheods of Operations Research, 2007, 66(3):409-418.
[20] R. Lucchetti, I. Revalski, (Eds.) Recent Development in Well-posed Variational Problems[M]. Kluwer Academic Publishers, Dordrecht, 1995.
[21] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness of constrained vector optimization problems[J]. Journal of Global Optimization, 2007, 37:287-304.
[22] Y. P. Fang, R. Hu, and N. J. Huang, Extended B-well-posedness and property (H) for set-valued vector optimization with convexity[J]. Journal of Optimization Theory and Applications, 2007, 135: 445–458.
[23] Y. Hu and Y. P. Fang, Set-valued increasing-along-rays maps and well-posed set-valued star-shaped optimization[J]. Journal of Mathematical Analysis and Applications, 2007, 331: 1371–1383.
[24] G. Y. Chen and X. Q. Yang, The vector complementarity problem and its equivalence with the weak minimal element in ordered spaces[J]. Journal of Mathematical Analysis and Applications, 1990, 153(l):136-156.
[25] F. Giannessi (Ed.), Vector Variational Inequalities and Vector Equilibria: Mathematical Theories[M]. Kluwer, Dordrecht, 2000.
[26] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness in generalized variational inequality problems with functional constraints[J]. Journal of Industrial and Management Optimization, 2007, 3(4):671-684.
[27] Y. P. Fang, R. Hu and N. J. Huang, Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints[J]. Computers and Mathematics with Applications, 2008, 55(1): 89-100.
[28] S. J. Li and M. H. Li, Levitin-Polyak well-posedness of vector equilibrium problems[J]. Mathematical Methods of Operations Research, 2009, 69: 125-140.
[29] M. B. Lignola, Well-posedness and L-well-posedness for quasivariational inequalities[J].Journal of Optimization Theory and Application, 2006, 128(1):119-138.
[30] M. Rocca, Well-posed vector optimization problems and vector variational inequalities[J]. Journal of Information and Optimization Sciences, 2006, 27(2):259--270.
[31] M. B. Lignola and J. Morgan, Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution[J]. Journal of Global Optimization, 2000, 16:57-67.
[32] Y. P. Fang and R. Hu, Parametric well-posedness for variational inequalities defined by bifunctions[J]. Computers and Mathematics with Applications, 2007, 53:1306-1316.
[33] G. P. Crespi, A. Guerraggio and M. Rocca, Well-posedness in vector optimization problems and vector variational inequalities[J]. Journal of Optimization Theory and Applications, 2007, 132(1):213-226.
[34] R. Ferrentino, Pointwise well-posedness in vector optimization and variational inequalities[R]. Working paper, Department of Economic Sciences and Statistics, University of Salerno-Fisciano, 2005.
[35] J. Yu, H. Yang and C. Yu, Well-posed Ky Fan’s point, quasi-variational inequality and Nash equilibrium problems[J]. Nonlinear Analysis, 2007, 66:777-790.
[36] M. Margiocco, F. Patrone and L. Pusillo Chicco, A new approach to Tykhonov well-posedness for Nash equilibria[J]. Optimization, 1997, 40:385-400.
[37] Y. P. Fang, and R. Hu, Estimates of approximate solutions and well-posedness for variational inequalities[J]. Mathematical Methods of Operations Research, 2007, 65:281–291.
[38] Y. P. Fang, N. J. Huang, and J. C. Yao, Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems[J]. Journal of Global Optimization, 2008, 41:117–133.
[39] L. C. Ceng, N. Hadjisavvas, S. Schaible, and J. C. Yao, Well-posedness for mixed quasivariational-like inequalities[J]. Journal of Optimization Theory and Applications, 2008, 139:109–125.
[40] L. C. Ceng, and J. C. Yao, Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems[J]. Nonlinear Analysis, 2008, 69:4585–4603.
[41] X. Zui, D. L. Zhu, and X. X. Huang, Levitin–Polyak well-posedness in generalized vector variational inequality problem with functional constraints[J]. Mathematical Methods of Operations Research, 2008, 67(3):505-524.
[42] L. J. Lin, and C. S. Chuang, Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraint[J]. Nonlinear Analysis, 2009, 70: 3609-3617.
[43] B. Jiang, J. Zhang, and X. X. Huang, Levitin–Polyak well-posedness of generalized quasivariational inequalities with functional constraints[J]. Nonlinear Analysis, 2009, 70:1492–1503.
[44] I. Del Prete, M. B. Lignola, and J. Morgan, New concepts of well-posedness for optimization problems with variational inequality constraint[J]. Journal of Inequalities in Pure and Applied Mathematics, 2002, 4(1), article 5.
[45] B. Lemarie, C. Ould Ahmed Salem, and J. P. Revalski, Well-posedness by perturbations of variational problems[J]. Journal of Optimization Theory and Applications, 2002, 115 (2):345–368.
[46] J. Morgan, and V. Scalzo, Discontinuous but well-posed optimization problems[J]. SIAM Journal on Optimization, 2006, 17(3):861–870.
[47] E. Blum, W. Oettli. From optimization and variational inequalities to equilibrium problems[J]. The Mathematics Student, 1994, 63: 123-145.
[48] M. Bianchi, N. Hadjisavvas, S. Schaible. Vector equilibrium problems with generalized monotone bifunctions[J]. Journal of Optimization Theory and Applications, 1997, 92: 527-542.
[49] Q. H. Ansari, W. Oettli, D. Schlager. A generalization of vectorial equilibria[J]. Mathematical Methods of Operations Research, 1997, 46: 147-152.
[50] G. Y. Chen, X. X. Huang, X. Q. Yang. Vector Optimization: Set-Valued and Variational Analysis[M]. Springer, Berlin, 2005.
[51] G. Y. Chen, C. J. Goh, X. Q. Yang. On gap functions for vector variational inequalities[A], in F. Giannessi (ed.), Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publishers, Dordrecht, Holland, 2000, 55-72.
[52] G. Y. Chen, S. J. Li. Existence of solutions for a generalized vector quasivariational inequality[J]. Journal of Optimization Theory and Applications, 1996, 90: 321-334.
[53] Q. H. Ansari, A. H. Siddiqi, S.Y. Wu. Existence and duality of generalized vector equilibrium problems[J]. Journal of Mathematical Analysis and Applications, 2001, 259: 115-126.
[54] X. H. Gong, J. C. Yao. Connectedness of the set of efficient solutions for generalized systems[J]. Journal of Optimization Theory and Applications, 2008, 138: 189-196.
[55] X. H. Gong. Connectedness of the solution sets and scalarization for vector equilibrium problems[J]. Journal of Optimization Theory and Applications, 2007, 133: 151-161.
[56] M. Margiocco, F. Patrone and L. Pusillo Chicco, Metric characterizations of Tykhonov well-posedness in value[J]. Journal of Optimization Theory and Applications, 1999, 100:377-387.
[57] H. Yang and J. Yu, Unified approaches to well-posedness with some applications[J]. Journal of Global Optimization, 2005, 31:371-381.
[58] A.V. Fiacco. Introduction to Sensitivity and Stablity Analysis in Nonlinear Programming[M]. Academic Press, New York, 1983.
[59] G. Y. Chen, B. D. Craven. Existence and continuity of solutions for vector optimization[J]. Journal of Optimization Theory and Applications, 1994, 81:459-468.
[60] R. E. Lucchetti, E. Miglierina. Stability for convex vector optimization problems[J]. Optimization, 2004, 53:517-528.
[61] T. Tanino, Y. Aawaragi. Stability of nondominated solutions in multicriteria decision- marking[J]. Journal of Optimization Theory and Applications, 1980, 30:229-253.
[62] Y. H. Cheng, D. L. Zhu. Global stability results for the weak vector variational inequality. Journal of Global Optimization, 2005, 32: 543-550.
[63] L. Q. Anh, P. Q. Khanh. Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems[J]. Journal of Mathematical Analysis and Applications, 2004, 294: 699-711.
[64] L. Q. Anh, P. Q. Khanh. On the stability of the solution sets of general multivalued vector quasiequilibrium problems[J]. Journal of Optimization Theory and Applications, 2007, 135: 271-284.
[65] L. Q. Anh, P. Q. Khanh. Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks I: Upper semicontinuities[J]. Set-Valued Analysis, 2008, 16: 267-279.
[66] L. Q. Anh, P. Q. Khanh. Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks II: Lower semicontinuities applications[J]. Set-Valued Analysis, 2008, 16: 943-960.
[67] N. J. Huang, J. Li, H. B. Thompson. Stability for parametric implicit vector equilibrium problems[J]. Mathematical and Computer Modelling, 2006, 43: 1267-1274.
[68] K. Kimura, J. C. Yao. Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems[J]. Journal of Global Optimization, 2008, 41: 187-202.
[69] X. H. Gong, J. C. Yao. Lower semicontinuity of the set of efficient solutions for generalized systems[J]. Journal of Optimization Theory and Applications, 2008, 138: 197-205.
[70] X. H. Gong. Continuity of the solution set to parametric weak vector equilibrium problems[J]. Journal of Optimization Theory and Applications, 2008, 139: 35-46.
[71] C. R. Chen, S. J. Li, K. L. Teo. Solution semicontinuity of parametric generalized vector equilibrium problems[J]. Journal of Global Optimization, 2009, 45: 309-318.
[72] S. J. Li, H. M. Liu, C. R. Chen. Lower semicontinuity of parametric generalized weak vector equilibrium problems[J]. Bulletin of the Australian Mathematical Society, 2010, 81: 85-95.
[73] D. Kuroiwa, Some duality theorems of set-valued optimization with natural criteria[A]. in: Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, World Scientific, River Edge, NJ, 1999, 221-228.
[74] D. Kuroiwa, On set-valued optimization[J]. Nonlinear Analysis, 2001, 47:1395-1400.
[75] D. Kuroiwa, Existence theorems of set optimization with set-valued maps[J]. Journal of Information and Optimization, 2003, 24:73-84.
[76] D. Kuroiwa, Existence of efficient points of set optimization with weighted criteria[J]. Journal of Nonlinear Convex Analysis, 2003, 4:117-123.
[77] T. X. D. Ha, Some variants of Ekeland variational priciple for a set-valued map[J]. Journal of Optimization Theory and Applications, 2005, 124(1):187-206.
[78] E. Hernández, L. Rodríguez-Marín, Nonconvex scalarization in set optimization with set-valued maps[J]. Journal of Mathematical Analysis and Applications, 2007, 325(1):1-18.
[79] M. Alonso, L. Rodríguez-Marín, Set-relations and optimality conditions in set-valued maps[J]. Nonlinear Analysis, 2005, 63:1167-1179.
[80] M. A. Noor and W. Oettli, On general nonlinear complementarity problems and quasi-equilibria[J]. Le Matematiche XLIX, 1994, 313-331 .
[81] J. Y. Fu, Symmetric vector quasi-equilibrium problems[J]. Journal of Mathematical Analysis and Applications, 2003, 285:708-713.
[82] A. P. Farajzadeh, On the symmetric vector quasi-equilibrium problems[J]. Journal of Mathematical Analysis and Applications, 2006, 2:1099-1110.
[83] R. E. Lucchetti and E. Miglierina, Stability for convex vector optimization problems[J]. Optimization, 2004, 53(5-6):517-528.
[84] T. Tanino, Stability and sensitivity analysis in convex vector optimization[J]. SIAM Journal on Control and Optimization, 1988, 26:521-536.
[85] H. Attouch, Variational Convergence for Functions an Operators[M]. Pitman, Boston, 1984.
[86] P. Oppezzi, A. M. Rossi, A convergence for vector-valued functions[J]. Optimization, 2008, 57(3):435-448.
[87] X. X. Huang, Stability in vector-valued and set-valued optimization[J]. Mathematical Metheods of Operations Research, 2000, 52:185-193.
[88] P. Oppezzi, A. M. Rossi, Existence and convergence of Pareto minima[J]. Journal of Optimization Theory and Applications, 2006, 128(3):653-664.
[89] A. G?pfort, H. Riahi, C. Tammer, C. Zǎlinescu, Variational methods in partially orderedspaces[M]. Springer, New York, 2003.
[90] S. Park, Fixed points and quasi-equilibrium problems[J]. Mathematical and Computer Modelling, 2001, 34:947-954.
[91] V. I. Istratescu, Fixed Point Theory, an Introduction[M]. Direidel Publishing Company, Dordrecht, 1981.
[92] D. T. Luc, Theory of Vector Optimization[M]. Springer-Verlag, Berlin, 1989.
[93] J. P. Aubin and H. Frankowska, Set-Valued Analysis[M]. Birkh?user, Boston, 1990.
[94] J. P. Aubin, and I. Ekeland, Applied Nonlinear Analysis[M]. John Wiley and Sons, New York, 1984.
[95] I. Ekeland, On the variational principle, Journal of Mathematical Analysis and Applications[J]. 1974, 47:324-353.
[96] S. J. Li, K. L. Teo, X. Q. Yang, and S. Y. Wu, Gap functions and existence of solutions to generalized vector quasi-equilibrium problems[J]. Journal of Global Optimization, 2006, 34:427-440.
[97] G. Y. Chen, X. Q. Yang, and H. Yu, A nonlinear scalarization function and generalized quasi-vector equilibrium problems[J]. Journal of Global Optimization, 2005, 32:451-466.
[98] C. Gutiérrez, B. Jiménez and V. Novo, On approximate solutions in vector optimization problems via scalarization[J]. Computational Optimization and Applications, 2006, 35:305-324.
[99] S. S. Kutateladze, Convexε-programming[J]. Soveit mathematics- Doklady, 1979, 20:391-393.
[100] R. T. Rockafellar, R. J.-B. Wets, Variational Analysis[M]. Springer-Verlag, Berlin, Heidelberg, 1998.
[101] M. Durea, On the existence and stability of approximate solutions of perturbed vector equilibrium problems[J]. Journal of Mathematical Analysis and Applications, 2007, 333:1165-1179.
[102] R. T. Rockafeller, Convex Analysis[M]. Princeton Univ. Press, Princeton, New Jersey, 1970.
[103] A. L. Peressini, Ordered Topological Vector Spaces[M]. Harper and Row, Now Yark, 1967.
[104] J.W. Nieuwenhuis, Some minimax theorems in vector-valued functions[J]. Journal of Optimization Theory and Applications, 1983,40:463-475.
[2] T. Zolezzi, Extended well-posedness of optimization problems[J]. Journal of Optimization Theory and Applications, 1996, 96:257-266.
[3] A. L. Dontchev, T. Zolezzi, Well-posed Optimization Problems[M]. In: Lecture Notes in Mathematics, 1543, Springer-Verlag, Berlin, 1993.
[4] E. S. Levitin, and B. T. Polyak, Convergence of minimizing sequences in conditional extremum problems[J]. Soviet Mathematics Doklady, 1966, 7:764-767.
[5] X. X. Huang, and X. Q. Yang, Generalized Levitin-Polyak well-posedness in constrained optimization[J]. SIAM Journal on Optimization, 2006, 17(1):243-258.
[6] E. Miglierina, and E. Molho, Well-posedness and stability for abstract spline problems[J]. Journal of Mathematical Analysis and Applications, 2007(333):1058–1069.
[7] T. Zolezzi, Well-posedness and optimization under perturbations[J]. Annals of Operations Research, 2001, 101:351-361.
[8] D. Dentcheva, S. Helbig, On variational principles, level sets, well-posedness andε-solutions in vector optimization[J]. Journal of Optimization Theory and Application, 1996, 89:329-349.
[9] X. X. Huang, Extended well-posedness Properties of vector optimization problems[J]. Journal of Optimization Theory and Applications, 2000, 106(1):165 -182.
[10] X. X. Huang, Extended and strongly extended well-posedness of set-valued optimization problems[J]. Mathematical Methods of Operations Research, 2001, 53:101-116.
[11] X. X. Huang, Pointwise well-posedness of perturbed vector optimization problems in a vector-valued variational principle[J]. Journal of Optimization Theory and Applications, 2001, 108(3):671-686.
[12] E. M. Bednarczuk, An approach to well-posedness in vector optimization problems: consequences to stability[J]. Control and Cybernetics, 1994, 23:107-122.
[13] E. Miglierina and E. Molho, Well-posedness and convexity in vector optimization[J]. Mathematical Methods Operations Reseach, 2003, 58:375-385.
[14] E. Miglierina, E. Molho and M. Rocca, Well-posedness and scalarization in vector optimization[J]. Journal of Optimization Theory and Applications, 2005, 126(2):391-409.
[15] R. Lucchetti, Well-posedness, towards vector optimization[J]. Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, 1987, 294:194-207.
[16] E. Bednarczuk, Well-posedness of vector optimization problems[J]. In: Jahn, J. and Krabs, W. (eds), Recent advances and historical development of vector optimization problems, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 1987, 294:51–61.
[17] P. Loridan, Well-posedness in vector optimization[A]. In: Lucchetti, R. and Revalski, J. (eds), Recent developments in variational well-posedness problems. Mathematics and its Applications, Kluwer Academic Publisher, Dordrecht, 1995, 331:171–192.
[18] Y. P. Fang and N. J. Huang, Increasing-along-rays property, vector optimization and well-posedness[J]. Mathematical Methods of Operations Research, 2007, 65: 99–114.
[19] M. Durea, Scalarization for pointwise well-posed vectorial problems[J]. Mathematical Metheods of Operations Research, 2007, 66(3):409-418.
[20] R. Lucchetti, I. Revalski, (Eds.) Recent Development in Well-posed Variational Problems[M]. Kluwer Academic Publishers, Dordrecht, 1995.
[21] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness of constrained vector optimization problems[J]. Journal of Global Optimization, 2007, 37:287-304.
[22] Y. P. Fang, R. Hu, and N. J. Huang, Extended B-well-posedness and property (H) for set-valued vector optimization with convexity[J]. Journal of Optimization Theory and Applications, 2007, 135: 445–458.
[23] Y. Hu and Y. P. Fang, Set-valued increasing-along-rays maps and well-posed set-valued star-shaped optimization[J]. Journal of Mathematical Analysis and Applications, 2007, 331: 1371–1383.
[24] G. Y. Chen and X. Q. Yang, The vector complementarity problem and its equivalence with the weak minimal element in ordered spaces[J]. Journal of Mathematical Analysis and Applications, 1990, 153(l):136-156.
[25] F. Giannessi (Ed.), Vector Variational Inequalities and Vector Equilibria: Mathematical Theories[M]. Kluwer, Dordrecht, 2000.
[26] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness in generalized variational inequality problems with functional constraints[J]. Journal of Industrial and Management Optimization, 2007, 3(4):671-684.
[27] Y. P. Fang, R. Hu and N. J. Huang, Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints[J]. Computers and Mathematics with Applications, 2008, 55(1): 89-100.
[28] S. J. Li and M. H. Li, Levitin-Polyak well-posedness of vector equilibrium problems[J]. Mathematical Methods of Operations Research, 2009, 69: 125-140.
[29] M. B. Lignola, Well-posedness and L-well-posedness for quasivariational inequalities[J].Journal of Optimization Theory and Application, 2006, 128(1):119-138.
[30] M. Rocca, Well-posed vector optimization problems and vector variational inequalities[J]. Journal of Information and Optimization Sciences, 2006, 27(2):259--270.
[31] M. B. Lignola and J. Morgan, Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution[J]. Journal of Global Optimization, 2000, 16:57-67.
[32] Y. P. Fang and R. Hu, Parametric well-posedness for variational inequalities defined by bifunctions[J]. Computers and Mathematics with Applications, 2007, 53:1306-1316.
[33] G. P. Crespi, A. Guerraggio and M. Rocca, Well-posedness in vector optimization problems and vector variational inequalities[J]. Journal of Optimization Theory and Applications, 2007, 132(1):213-226.
[34] R. Ferrentino, Pointwise well-posedness in vector optimization and variational inequalities[R]. Working paper, Department of Economic Sciences and Statistics, University of Salerno-Fisciano, 2005.
[35] J. Yu, H. Yang and C. Yu, Well-posed Ky Fan’s point, quasi-variational inequality and Nash equilibrium problems[J]. Nonlinear Analysis, 2007, 66:777-790.
[36] M. Margiocco, F. Patrone and L. Pusillo Chicco, A new approach to Tykhonov well-posedness for Nash equilibria[J]. Optimization, 1997, 40:385-400.
[37] Y. P. Fang, and R. Hu, Estimates of approximate solutions and well-posedness for variational inequalities[J]. Mathematical Methods of Operations Research, 2007, 65:281–291.
[38] Y. P. Fang, N. J. Huang, and J. C. Yao, Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems[J]. Journal of Global Optimization, 2008, 41:117–133.
[39] L. C. Ceng, N. Hadjisavvas, S. Schaible, and J. C. Yao, Well-posedness for mixed quasivariational-like inequalities[J]. Journal of Optimization Theory and Applications, 2008, 139:109–125.
[40] L. C. Ceng, and J. C. Yao, Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems[J]. Nonlinear Analysis, 2008, 69:4585–4603.
[41] X. Zui, D. L. Zhu, and X. X. Huang, Levitin–Polyak well-posedness in generalized vector variational inequality problem with functional constraints[J]. Mathematical Methods of Operations Research, 2008, 67(3):505-524.
[42] L. J. Lin, and C. S. Chuang, Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraint[J]. Nonlinear Analysis, 2009, 70: 3609-3617.
[43] B. Jiang, J. Zhang, and X. X. Huang, Levitin–Polyak well-posedness of generalized quasivariational inequalities with functional constraints[J]. Nonlinear Analysis, 2009, 70:1492–1503.
[44] I. Del Prete, M. B. Lignola, and J. Morgan, New concepts of well-posedness for optimization problems with variational inequality constraint[J]. Journal of Inequalities in Pure and Applied Mathematics, 2002, 4(1), article 5.
[45] B. Lemarie, C. Ould Ahmed Salem, and J. P. Revalski, Well-posedness by perturbations of variational problems[J]. Journal of Optimization Theory and Applications, 2002, 115 (2):345–368.
[46] J. Morgan, and V. Scalzo, Discontinuous but well-posed optimization problems[J]. SIAM Journal on Optimization, 2006, 17(3):861–870.
[47] E. Blum, W. Oettli. From optimization and variational inequalities to equilibrium problems[J]. The Mathematics Student, 1994, 63: 123-145.
[48] M. Bianchi, N. Hadjisavvas, S. Schaible. Vector equilibrium problems with generalized monotone bifunctions[J]. Journal of Optimization Theory and Applications, 1997, 92: 527-542.
[49] Q. H. Ansari, W. Oettli, D. Schlager. A generalization of vectorial equilibria[J]. Mathematical Methods of Operations Research, 1997, 46: 147-152.
[50] G. Y. Chen, X. X. Huang, X. Q. Yang. Vector Optimization: Set-Valued and Variational Analysis[M]. Springer, Berlin, 2005.
[51] G. Y. Chen, C. J. Goh, X. Q. Yang. On gap functions for vector variational inequalities[A], in F. Giannessi (ed.), Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publishers, Dordrecht, Holland, 2000, 55-72.
[52] G. Y. Chen, S. J. Li. Existence of solutions for a generalized vector quasivariational inequality[J]. Journal of Optimization Theory and Applications, 1996, 90: 321-334.
[53] Q. H. Ansari, A. H. Siddiqi, S.Y. Wu. Existence and duality of generalized vector equilibrium problems[J]. Journal of Mathematical Analysis and Applications, 2001, 259: 115-126.
[54] X. H. Gong, J. C. Yao. Connectedness of the set of efficient solutions for generalized systems[J]. Journal of Optimization Theory and Applications, 2008, 138: 189-196.
[55] X. H. Gong. Connectedness of the solution sets and scalarization for vector equilibrium problems[J]. Journal of Optimization Theory and Applications, 2007, 133: 151-161.
[56] M. Margiocco, F. Patrone and L. Pusillo Chicco, Metric characterizations of Tykhonov well-posedness in value[J]. Journal of Optimization Theory and Applications, 1999, 100:377-387.
[57] H. Yang and J. Yu, Unified approaches to well-posedness with some applications[J]. Journal of Global Optimization, 2005, 31:371-381.
[58] A.V. Fiacco. Introduction to Sensitivity and Stablity Analysis in Nonlinear Programming[M]. Academic Press, New York, 1983.
[59] G. Y. Chen, B. D. Craven. Existence and continuity of solutions for vector optimization[J]. Journal of Optimization Theory and Applications, 1994, 81:459-468.
[60] R. E. Lucchetti, E. Miglierina. Stability for convex vector optimization problems[J]. Optimization, 2004, 53:517-528.
[61] T. Tanino, Y. Aawaragi. Stability of nondominated solutions in multicriteria decision- marking[J]. Journal of Optimization Theory and Applications, 1980, 30:229-253.
[62] Y. H. Cheng, D. L. Zhu. Global stability results for the weak vector variational inequality. Journal of Global Optimization, 2005, 32: 543-550.
[63] L. Q. Anh, P. Q. Khanh. Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems[J]. Journal of Mathematical Analysis and Applications, 2004, 294: 699-711.
[64] L. Q. Anh, P. Q. Khanh. On the stability of the solution sets of general multivalued vector quasiequilibrium problems[J]. Journal of Optimization Theory and Applications, 2007, 135: 271-284.
[65] L. Q. Anh, P. Q. Khanh. Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks I: Upper semicontinuities[J]. Set-Valued Analysis, 2008, 16: 267-279.
[66] L. Q. Anh, P. Q. Khanh. Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks II: Lower semicontinuities applications[J]. Set-Valued Analysis, 2008, 16: 943-960.
[67] N. J. Huang, J. Li, H. B. Thompson. Stability for parametric implicit vector equilibrium problems[J]. Mathematical and Computer Modelling, 2006, 43: 1267-1274.
[68] K. Kimura, J. C. Yao. Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems[J]. Journal of Global Optimization, 2008, 41: 187-202.
[69] X. H. Gong, J. C. Yao. Lower semicontinuity of the set of efficient solutions for generalized systems[J]. Journal of Optimization Theory and Applications, 2008, 138: 197-205.
[70] X. H. Gong. Continuity of the solution set to parametric weak vector equilibrium problems[J]. Journal of Optimization Theory and Applications, 2008, 139: 35-46.
[71] C. R. Chen, S. J. Li, K. L. Teo. Solution semicontinuity of parametric generalized vector equilibrium problems[J]. Journal of Global Optimization, 2009, 45: 309-318.
[72] S. J. Li, H. M. Liu, C. R. Chen. Lower semicontinuity of parametric generalized weak vector equilibrium problems[J]. Bulletin of the Australian Mathematical Society, 2010, 81: 85-95.
[73] D. Kuroiwa, Some duality theorems of set-valued optimization with natural criteria[A]. in: Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, World Scientific, River Edge, NJ, 1999, 221-228.
[74] D. Kuroiwa, On set-valued optimization[J]. Nonlinear Analysis, 2001, 47:1395-1400.
[75] D. Kuroiwa, Existence theorems of set optimization with set-valued maps[J]. Journal of Information and Optimization, 2003, 24:73-84.
[76] D. Kuroiwa, Existence of efficient points of set optimization with weighted criteria[J]. Journal of Nonlinear Convex Analysis, 2003, 4:117-123.
[77] T. X. D. Ha, Some variants of Ekeland variational priciple for a set-valued map[J]. Journal of Optimization Theory and Applications, 2005, 124(1):187-206.
[78] E. Hernández, L. Rodríguez-Marín, Nonconvex scalarization in set optimization with set-valued maps[J]. Journal of Mathematical Analysis and Applications, 2007, 325(1):1-18.
[79] M. Alonso, L. Rodríguez-Marín, Set-relations and optimality conditions in set-valued maps[J]. Nonlinear Analysis, 2005, 63:1167-1179.
[80] M. A. Noor and W. Oettli, On general nonlinear complementarity problems and quasi-equilibria[J]. Le Matematiche XLIX, 1994, 313-331 .
[81] J. Y. Fu, Symmetric vector quasi-equilibrium problems[J]. Journal of Mathematical Analysis and Applications, 2003, 285:708-713.
[82] A. P. Farajzadeh, On the symmetric vector quasi-equilibrium problems[J]. Journal of Mathematical Analysis and Applications, 2006, 2:1099-1110.
[83] R. E. Lucchetti and E. Miglierina, Stability for convex vector optimization problems[J]. Optimization, 2004, 53(5-6):517-528.
[84] T. Tanino, Stability and sensitivity analysis in convex vector optimization[J]. SIAM Journal on Control and Optimization, 1988, 26:521-536.
[85] H. Attouch, Variational Convergence for Functions an Operators[M]. Pitman, Boston, 1984.
[86] P. Oppezzi, A. M. Rossi, A convergence for vector-valued functions[J]. Optimization, 2008, 57(3):435-448.
[87] X. X. Huang, Stability in vector-valued and set-valued optimization[J]. Mathematical Metheods of Operations Research, 2000, 52:185-193.
[88] P. Oppezzi, A. M. Rossi, Existence and convergence of Pareto minima[J]. Journal of Optimization Theory and Applications, 2006, 128(3):653-664.
[89] A. G?pfort, H. Riahi, C. Tammer, C. Zǎlinescu, Variational methods in partially orderedspaces[M]. Springer, New York, 2003.
[90] S. Park, Fixed points and quasi-equilibrium problems[J]. Mathematical and Computer Modelling, 2001, 34:947-954.
[91] V. I. Istratescu, Fixed Point Theory, an Introduction[M]. Direidel Publishing Company, Dordrecht, 1981.
[92] D. T. Luc, Theory of Vector Optimization[M]. Springer-Verlag, Berlin, 1989.
[93] J. P. Aubin and H. Frankowska, Set-Valued Analysis[M]. Birkh?user, Boston, 1990.
[94] J. P. Aubin, and I. Ekeland, Applied Nonlinear Analysis[M]. John Wiley and Sons, New York, 1984.
[95] I. Ekeland, On the variational principle, Journal of Mathematical Analysis and Applications[J]. 1974, 47:324-353.
[96] S. J. Li, K. L. Teo, X. Q. Yang, and S. Y. Wu, Gap functions and existence of solutions to generalized vector quasi-equilibrium problems[J]. Journal of Global Optimization, 2006, 34:427-440.
[97] G. Y. Chen, X. Q. Yang, and H. Yu, A nonlinear scalarization function and generalized quasi-vector equilibrium problems[J]. Journal of Global Optimization, 2005, 32:451-466.
[98] C. Gutiérrez, B. Jiménez and V. Novo, On approximate solutions in vector optimization problems via scalarization[J]. Computational Optimization and Applications, 2006, 35:305-324.
[99] S. S. Kutateladze, Convexε-programming[J]. Soveit mathematics- Doklady, 1979, 20:391-393.
[100] R. T. Rockafellar, R. J.-B. Wets, Variational Analysis[M]. Springer-Verlag, Berlin, Heidelberg, 1998.
[101] M. Durea, On the existence and stability of approximate solutions of perturbed vector equilibrium problems[J]. Journal of Mathematical Analysis and Applications, 2007, 333:1165-1179.
[102] R. T. Rockafeller, Convex Analysis[M]. Princeton Univ. Press, Princeton, New Jersey, 1970.
[103] A. L. Peressini, Ordered Topological Vector Spaces[M]. Harper and Row, Now Yark, 1967.
[104] J.W. Nieuwenhuis, Some minimax theorems in vector-valued functions[J]. Journal of Optimization Theory and Applications, 1983,40:463-475.