- Research Article
- Open Access
Levitin-Polyak Well-Posedness in Vector Quasivariational Inequality Problems with Functional Constraints
© J. Zhang et al. 2010
Received: 17 March 2010
Accepted: 6 July 2010
Published: 20 July 2010
We introduce several types of Levtin-Polyak well-posedness for a vector quasivariational inequality with functional constraints. Necessary and/or sufficient conditions are derived for them.
It is well known that, under certain conditions, a Nash equilibrium problem can be formulated and solved as a variational inequality problem. A generalized Nash game is a Nash game in which each player's strategy depends on other players' strategies . The connection between generalized Nash games and quasivariational inequalities was first recognized by Bensoussan . Recently, some researchers [1, 3, 4] found that mathematical models of many real world problems, including some engineering problems, can be formulated as certain kinds of variational inequality problems, including quasivariational inequality problems. However, as noted in , compared with variational inequality problems, the study on quasivariational inequality problems is still in its infancy, in particular only a few algorithms have been proposed to solve variational inequalities numerically.
Vector variational inequality problems were introduced by Giannessi  and are related to vector network equilibrium problems . Since then, various types of vector variational inequalities were introduced and studied (see, e.g., [8, 9] and the references therein).
In this paper, we will consider vector quasivariational inequality problems with functional constraints, which are described below.
Well-posedness for unconstrained and constrained optimization problems was first studied by Tikhonov  and Levitin and Polyak . The issue being considered is that for each approximating solution sequence, there exists a subsequence that converges to a solution of the problem.
In Tikhonov's well posedness, the approximating solution is always feasible. However, it should be noted that many algorithms in optimization and variational inequalities, such as penalty-type methods and augmented Lagrangian methods, terminate when the constraint is approximately satisfied. These methods may generate sequences that may not be necessarily feasible .
Up to now, various extensions of these well posednesses have been developed and well studied (see, e.g., [13–18]). Studies on well posedness of optimization problems have been extended to vector optimization problems (see e.g., [19–24]). The study of Levitin-Polyak well posedness for scalar convex optimization problems with functional constraints originates from . Recently, this research was extended to nonconvex optimization problems with abstract and functional constraints  and nonconvex vector optimization problems with both abstract and functional constraints . Well-posedness of variational inequality problems, mixed variational inequality problems, and equilibrium problems without functional constraints was investigated in the literature (see, e.g., [27–30]). Well-posedness in variational inequality problems with both abstract and functional constraints was investigated in . Well-posedness of (generalized) quasivariational inequality and mixed quasivariational-like inequalities has been studied in the literature [32–35]. The study of well posedness for (generalized) vector variational inequality, vector quasiequilibria and vector equilibrium problems can be found in [36–39] and the references therein.
In this paper, we will introduce and study several types of Levitin-Polyak (LP in short) well posednesses and generalized LP well posednesses for vector quasivariational inequalities with functional constraints. The paper is organized as follows. In Section 2, four types of LP well posednesses and generalized LP well posednesses for vector quasivariational inequality problems will be defined. In Section 3, we will derive various criteria and characterizations for the various (generalized) LP well posednesses of constrained vector quasivariational inequalities.
2. Definitions and Preliminaries
(VQVI) is said to be type I (resp., type II, generalized type I, generalized type II) LP well posed if the solution set of (VQVI) is nonempty, and for any type I (resp., type II, generalized type I, generalized type II) LP approximating solution sequence , there exist a subsequence of and such that .
It is easily seen that if , , then type I (resp., type II, generalized type I, generalized type II) LP well posedness of (VQVI) reduces to type I (resp., type II, generalized type I, generalized type II) LP well posedness of (QVI) defined in .
It is clear that any (generalized) type II LP approximating solution sequence is a (generalized) type I LP approximating solution sequence. Thus, (generalized) type I LP well posedness implies (generalized) type II LP well posedness.
then . In this paper, we always assume that . We note that LP well posedness for the special case, where is finite valued and l.s.c., is closed, has been studied in .
First we have the following lemma.
The following lemma establishes some relationship between LP approximating solution sequence and LP minimizing sequence.
Similar to the proof of Lemma 2.7, it is also routine to check that a sequence is a generalized type I (resp., generalized type II) LP approximating solution sequence if and only if it is a generalized type I (resp., generalized type II) LP minimizing sequence of (P). So (VQVI) is generalized type I (resp., generalized type II) LP well posed if and only if (P) is generalized type I (resp., generalized type II) LP well posed with defined by (2.26).
Since , for any . This fact together with Lemma 2.7 implies that a type I (resp., type II) LP minimizing sequence of (P) is a type I (resp., type II) LP approximating solution sequence. So type I (resp., type II) LP well posedness of (VQVI) implies type I (resp., type II) LP well posedness of (P) with defined by (2.26).
To end this section, we note that all the results in  for the well posedness hold for (P) so long as is closed, is l.s.c. on , and .
3. Criteria and Characterizations for Various LP Well-Posedness of (VQVI)
In this section, we give necessary and/or sufficient conditions for the various types of (generalized) LP well posednesses defined in Section 2.
The next proposition can be straightforwardly proved.
If (VQVI) is type I (resp., type II, generalized type I, generalized type II) LP well posed, then (3.1) holds. Conversely, if (3.1) holds and is compact, then (VQVI) is type I (resp., type II, generalized type I, generalized type II) LP well posed.
With the help of Lemma 2.7, analogously to [35, Theorems , and ], we can prove the following two theorems.
Next we give Furi-Vignoli type characterizations  for the (generalized) type I LP well posednesses of (VQVI).
we have and . We will show that , for all . Otherwise, there exists such that . By the weak* compactness of , we have , which leads to and gives rise to a contradiction. Furthermore, we observe that . This fact combined with implies that .
The next lemma can be proved analogously to ([25, Theorem ]).
To continue our study, we make some assumptions below.
Note that the function defined by (2.14) is nonnegative on . By the lower semicontinuity of and Lemma 3.6, is l.s.c. on . Moreover, is closed, since is closed on . By Proposition 2.8, Lemmas 3.4 and 3.5, the conclusion follows.
Although the type I (type II) LP well posedness of (VQVI) is not equivalent to the type I (type II) LP well posedness of (P), we can still establish the same characterization for type I (type II) LP well posedness of (VQVI) as Theorem 3.7. We need the next lemma.
The proof is similar to that of Theorem in  and thus omitted.
with , obviously is still closed and l.s.c. If we replace by in (i), then with , which is unbounded. Therefore, and the (VQVI) is not LP well posed in sense of type I. Actually, the solution set of this problem is and thus unbounded.
The following proposition presents some sufficient conditions for type I LP well posedness of (VQVI)
Let Assumption 1 hold. Further assume that one of the following conditions holds.
which contradicts condition (3.27).
Similarly, we can prove the next proposition.
Let Assumption 1 hold. Further assume that one of the following conditions holds.
The next proposition follows immediately from Proposition 2.8(i), Lemma 3.6, and [12, Proposition (iv)].
Let be a normed space, let be a closed and convex cone with nonempty interior and . Let the set-valued map be closed and l.s.c on . Assume that the solution set of (VQVI) is nonempty. Further assume that there exists such that the function defined by (2.14) is level compact on , then (VQVI) is generalized type I LP well posed.
This work is supported by the National Science Foundation of China.
- Facchinei F, Pang J-S: Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research. Volume 1–2. Springer, New York, NY, USA; 2003:xxxiv+624+I69.MATHGoogle Scholar
- Bensoussan A: Points de nash dans le cas de fonctionnelles quadratiques et jeux différentiels linéaires à personnes. SIAM Journal on Control and Optimization 1974, 12: 460–499. 10.1137/0312037MathSciNetView ArticleMATHGoogle Scholar
- Pang J-S, Fukushima M: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Computational Management Science 2005,2(1):21–56. 10.1007/s10287-004-0010-0MathSciNetView ArticleMATHGoogle Scholar
- Baiocchi C, Capelo A: Variational and Quasivariational Inequalities, A Wiley-Interscience Publication. John Wiley & Sons, New York, NY, USA; 1984:ix+452.Google Scholar
- Fukushima M: A class of gap functions for quasi-variational inequality problems. Journal of Industrial and Management Optimization 2007,3(2):165–171.MathSciNetView ArticleMATHGoogle Scholar
- Giannessi F: Theorems of alternative, quadratic programs and complementarity problems. In Variational Inequalities and Complementarity Problems. Edited by: Cottle RW, Giannessi F, Lions JL. John Wiley & Sons, New York, NY, USA; 1980:151–186.Google Scholar
- Chen GY, Yen ND: On the variational inequality model for network equilibrium. In Internal Report. Department of Mathematics, University of Pisa, Pisa, Italy; 1993.Google Scholar
- Ceng LC, Chen GY, Huang XX, Yao J-C: Existence theorems for generalized vector variational inequalities with pseudomonotonicity and their applications. Taiwanese Journal of Mathematics 2008,12(1):151–172.MathSciNetMATHGoogle Scholar
- Chen GY, Huang XX, Yang XQ: Vector Optimization, Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems. Volume 541. Springe, Berlin, Germany; 2005:x+306.MATHGoogle Scholar
- Tikhonov AN: On the stability of the functional optimization problem. USSR Computational Mathematics and Mathematical Physics 1966,6(4):28–33. 10.1016/0041-5553(66)90003-6View ArticleMATHGoogle Scholar
- Levitin ES, Polyak BT: Convergence of minimizing sequences in conditional extremum problems. Soviet Mathematics Doklady 1966, 7: 764–767.MATHGoogle Scholar
- Huang XX, Yang XQ: Generalized Levitin-Polyak well-posedness in constrained optimization. SIAM Journal on Optimization 2006,17(1):243–258. 10.1137/040614943MathSciNetView ArticleMATHGoogle Scholar
- Beer G, Lucchetti R: The epi-distance topology: continuity and stability results with applications to convex optimization problems. Mathematics of Operations Research 1992,17(3):715–726. 10.1287/moor.17.3.715MathSciNetView ArticleMATHGoogle Scholar
- Dontchev AL, Zolezzi T: Well-Posed Optimization Problems, Lecture Notes in Mathematics. Volume 1543. Springer, Berlin, Germany; 1993:xii+421.MATHGoogle Scholar
- Revalski JP: Hadamard and strong well-posedness for convex programs. SIAM Journal on Optimization 1997,7(2):519–526. 10.1137/S1052623495286776MathSciNetView ArticleMATHGoogle Scholar
- Lucchetti R, Revalski J (Eds): Recent Developments in Well-Posed Variational Problems, Mathematics and its Applications. Volume 331. Kluwer Academic, Dodrecht, The Netherlands; 1995:viii+266.MATHGoogle Scholar
- Zolezzi T: Well-posedness criteria in optimization with application to the calculus of variations. Nonlinear Analysis: Theory, Methods & Applications 1995,25(5):437–453. 10.1016/0362-546X(94)00142-5MathSciNetView ArticleMATHGoogle Scholar
- Zolezzi T: Extended well-posedness of optimization problems. Journal of Optimization Theory and Applications 1996,91(1):257–266. 10.1007/BF02192292MathSciNetView ArticleMATHGoogle Scholar
- Bednarczuk E: Well-posedness of vector optimization problems. In Recent Advances and Historical Development of Vector Optimization Problems, Lecture Notes in Economics and Mathematical Systems. Volume 294. Edited by: Jahn J, Krabs W. Springer, Berlin, Germany; 1987:51–61.View ArticleGoogle Scholar
- Deng S: Coercivity properties and well-posedness in vector optimization. RAIRO Operations Research 2003,37(3):195–208. 10.1051/ro:2003021MathSciNetView ArticleMATHGoogle Scholar
- Huang XX: Extended well-posedness properties of vector optimization problems. Journal of Optimization Theory and Applications 2000,106(1):165–182. 10.1023/A:1004615325743MathSciNetView ArticleMATHGoogle Scholar
- Huang XX: Extended and strongly extended well-posedness of set-valued optimization problems. Mathematical Methods of Operations Research 2001,53(1):101–116. 10.1007/s001860000100MathSciNetView ArticleMATHGoogle Scholar
- Loridan P: Well-posedness in vector optimization. In Recent Developments in Well-Posed Variational Problems, Mathematics and Its Applications. Volume 331. Kluwer Academic, Dodrecht, The Netherlands; 1995:171–192.View ArticleGoogle Scholar
- Lucchetti R: Well-Posedness Towards Vector Optimization, Lecture Notes in Economics and Mathematical System. Volume 294. Springer, New York, NY, USA; 1987.Google Scholar
- Konsulova AS, Revalski JP: Constrained convex optimization problems-well-posedness and stability. Numerical Functional Analysis and Optimization 1994,15(7–8):889–907. 10.1080/01630569408816598MathSciNetView ArticleMATHGoogle Scholar
- Huang XX, Yang XQ: Levitin-Polyak well-posedness of constrained vector optimization problems. Journal of Global Optimization 2007,37(2):287–304. 10.1007/s10898-006-9050-zMathSciNetView ArticleMATHGoogle Scholar
- Fang Y-P, Huang N-J, Yao J-C: Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems. Journal of Global Optimization 2008,41(1):117–133. 10.1007/s10898-007-9169-6MathSciNetView ArticleMATHGoogle Scholar
- Fang Y-P, Huang N-J, Yao J-C: Well-posedness by perturbations of mixed variational inequalities in Banach spaces. European Journal of Operational Research 2010,201(3):682–692. 10.1016/j.ejor.2009.04.001MathSciNetView ArticleMATHGoogle Scholar
- Lignola MB, Morgan J: -well-posedness for Nash equilibria and for optimization problems with Nash equilibrium constraints. Journal of Global Optimization 2006,36(3):439–459. 10.1007/s10898-006-9020-5MathSciNetView ArticleMATHGoogle Scholar
- Lignola MB, Morgan J: Generalized variational inequalities with pseudomonotone operators under perturbations. Journal of Optimization Theory and Applications 1999,101(1):213–220. 10.1023/A:1021783313936MathSciNetView ArticleMATHGoogle Scholar
- Huang XX, Yang XQ, Zhu DL: Levitin-Polyak well-posedness of variational inequality problems with functional constraints. Journal of Global Optimization 2009,44(2):159–174. 10.1007/s10898-008-9310-1MathSciNetView ArticleMATHGoogle Scholar
- Ceng LC, Hadjisavvas N, Schaible S, Yao JC: Well-posedness for mixed quasivariational-like inequalities. Journal of Optimization Theory and Applications 2008,139(1):109–125. 10.1007/s10957-008-9428-9MathSciNetView ArticleMATHGoogle Scholar
- Lignola MB: Well-posedness and -well-posedness for quasivariational inequalities. Journal of Optimization Theory and Applications 2006,128(1):119–138. 10.1007/s10957-005-7561-2MathSciNetView ArticleMATHGoogle Scholar
- Huang XX, Jiang B, Zhang J: Levitin-Polyak well-posedness of quasivariational inequality problems with functional constraints. Natural Science Jourmal of Xiangtan University 2008, 30: 1–11.MATHGoogle Scholar
- Jiang B, Zhang J, Huang XX: Levitin-Polyak well-posedness of generalized quasivariational inequalities with functional constraints. Nonlinear Analysis: Theory, Methods & Applications 2009,70(4):1492–1503. 10.1016/j.na.2008.02.029MathSciNetView ArticleMATHGoogle Scholar
- Anh LQ, Khanh PQ, Van DTM, Yao J-C: Well-posedness for vector quasiequilibria. Taiwanese Journal of Mathematics 2009,13(2):713–737.MathSciNetMATHGoogle Scholar
- Huang XX, Yang XQ: Levitin-Polyak well-posedness of vector variational inequality problems with functional constraints. Numerical Functional Analysis and Optimization 2010,31(4):440–459. 10.1080/01630563.2010.485296MathSciNetView ArticleMATHGoogle Scholar
- Li SJ, Li MH: Levitin-Polyak well-posedness of vector equilibrium problems. Mathematical Methods of Operations Research 2009,69(1):125–140. 10.1007/s00186-008-0214-0MathSciNetView ArticleMATHGoogle Scholar
- Wang G, Huang XX, Zhang J, Chen GY: Levitin-Polyak well-posedness for vector equilibrium problems with functional constraints. Acta Mathematica Scientia 2010,30(5):1400–1412. 10.1016/S0252-9602(10)60132-4MathSciNetView ArticleMATHGoogle Scholar
- Huang XX, Teo KL, Yang XQ: Calmness and exact penalization in vector optimization with cone constraints. Computational Optimization and Applications 2006,35(1):47–67. 10.1007/s10589-006-6441-5MathSciNetView ArticleMATHGoogle Scholar
- Furi M, Vignoli A: About well-posed minimization problems for functionals in metric spaces. Journal of Optimization Theory and Applications 1970, 5: 225–229. 10.1007/BF00927717View ArticleMATHGoogle Scholar
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