Research Article | Open Access

# LP Well-Posedness for Bilevel Vector Equilibrium and Optimization Problems with Equilibrium Constraints

**Academic Editor:**Chong Li

#### Abstract

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.

#### 1. Introduction

Well-posedness is one of most important topics for optimization theory and numerical methods of optimization problems, which guarantees that, for approximating solution sequences, there is a subsequence which converges to a solution. The well-posedness of unconstrained and constrained scalar optimization problems was first introduced and studied by Tikhonov [1] and Levitin and Polyak [2], respectively. In Tikhonov well-posedness, which means the existence and uniqueness of minimizer and the convergence of a subsequence of each approximation sequence to a solution, the Tikhonov notation has been more interested; that is, any algorithm can generate only an approximating sequence of solutions. Hence, this sequence is applicable only if the problem under consideration is well posed. The concept of Tikhonov well-posedness has also been generalized to several related problems: variational inequalities, Nash equilibrium problems, optimization problems with variational inequalities constrains, optimization problems with Nash equilibrium constrains, optimization problems with Nash equilibrium constrains [3–12], and so forth.

The study of Levitin-Polyak (LP for short) well-posedness for scalar convex optimization problems with functional constraints originates from [13]. Recently, this research was extended to nonconvex optimization problems with abstract and functional constraints [14] and nonconvex vector optimization problems with both abstract and functional constraints [15]. In 2009, S. J. Li and M. H. Li [16] introduced and researched two types of LP well-posedness of vector equilibrium problems with variable domination structures. In the same year, Huang et al. [17] introduced and researched the LP well-posedness of vector quasi-equilibrium problems. Moreover, Li et al. [18] introduced and researched the LP well-posedness for two types of generalized vector quasi-equilibrium problems.

Most recently, many papers appeared dealing with bilevel problems such as mathematical programming with equilibrium constraints [19, 20], optimization problems with Nash equilibrium constraints [21], optimization problems with variational inequality constraints [4], and optimization problems with equilibrium constraints [20, 22]. In 2012, Anh et al. [23] considered the LP well-posedness of bilevel equilibrium problems with equilibrium constraints (BEPEC) and bilevel optimization problems with equilibrium constraints (BOPEC). They introduced a relaxed level closedness and use it together with pseudomonotonic assumptions to establish sufficient conditions of LP well-posedness.

The vector equilibrium problem is a unified model of several classes of problems, for example, vector optimization problems and vector variational inequality problems [24, 25]. In recent years, many authors have intensively studied different types of vector equilibrium problem [26–28]. Many results on existence and stability of solutions for vector equilibrium problem, generalized vector equilibrium problems, and generalized quasivector equilibrium problems have been established [26–31]. Moreover, many authors have investigated the gap functions for vector equilibrium problems, generalized vector equilibrium problems, and generalized quasivector equilibrium problems [28].

Motivated and inspired by the above observations, our consideration of LP well-posedness for bilevel problems is in this paper. We focus on vector equilibria with equilibrium constraints and optimization with equilibrium constraints, as well as an abstract set constraint, and investigate criteria and characterizations for these types of LP well-posedness with a gap function for bilevel vector equilibrium problems with equilibrium constraints and optimization problems with equilibrium constraints. We propose a generalized level closedness and use it together to study well-posedness in the LP sense. However, since the existence topic has been intensively studied for vector equilibrium and bilevel problems, we focus on LP well-posedness, assuming always that the mentioned solutions exist.

The layout of the paper is as follows. In Section 2, we state the bilevel problems under our consideration and recall notions and preliminaries needed in the sequel. In Section 3, we study LP well-posedness of bilevel vector equilibrium problems with equilibrium constraints and optimization problems with equilibrium constraints on diameters and measures of noncompactness of approximate solution sets in the Kuratowski, Hausdorff, or Istrǎtescu sense. In Section 4, by virtue of a nonlinear scalarization function and a gap function for bilevel vector quasi-equilibrium problems, we show equivalent relations between the LP well-posedness of the optimization problem and the LP well-posedness of bilevel vector equilibrium problems. The results in this paper unify, generalize, and extend some known results in [16, 23].

#### 2. Preliminaries

Let and be locally convex Hausdorff topological vector spaces, where is a metric which is compatible with the topology of . Throughout this paper, suppose is a set-valued mapping such that, for any , is a pointed, closed, and convex cone in with nonempty interior , where depends on . We also assume that is a continuous vector-valued mapping and satisfies that, for any , . Let be a vector-valued mapping and , . The constraints appear in this paper are solution sets of the following (parametric) vector quasi-equilibrium problem, for each : Instead of writing for the family of quasi-equilibrium problems, that is, the parametric problem, we will simply write (VQEP) in the sequel. Let be the solution map of (VQEP). Let , , and be two functions with . We consider the following bilevel vector equilibrium problem with equilibrium constraints: where denotes the graph of ; that is, . We denote by the set of solutions of (BVEPEC).

We first defined LP well-posedness notions.

*Definition 1. *A sequence is called type I LP approximating sequence for (BVEPEC) if and only if there exists a sequence of nonnegative real number with such that

*Definition 2. *A sequence is called type II LP approximating sequence for (BVEPEC) if and only if there exists a sequence of nonnegative real number with such that (3)–(5) hold and for any there exists such that

*Definition 3. *Problems (BVEPEC) is called type I (resp., type II) LP well-posed if and only if(i)the solution set of (BVEPEC) is nonempty;(ii)for any type I (resp., type II), LP approximating sequence of (BVEPEC) has a subsequence converging to a solution.

Recall now some notions. Let and be as above and let be a multifunction. is called lower semicontinuous (lsc) at if and only if . For some open subsets, implies the existence of a neighborhood of such that for all . is upper semicontinuous (usc) at if and only if, for each open subset , there is a neighborhood of such that . is called closed at if and only if, for each net with , one has . We say that satisfies a certain property in a subset if and only if satisfies it at every point of . If , we omit “in ” in the saying. The following assertions are known:(i) is lsc at if and only if , , , and ;(ii) is closed if and only if is closed;(iii) is usc at if is compact for any compact subset of and is closed at ;(iv) is usc at if is compact and is closed at .

*Definition 4 (see [23]). *Let and be topological spaces, and . (i) is called upper -level closed at if and only if, for any sequence converging to ,
(ii) is called lower -level closed at if and only if, for any sequence converging to ,

*Definition 5. *Let be Hausdorff topological spaces and let be a topological vector space . The function is called upper -level closed at if and only if, for any converging to ,

*Definition 6 (see [32]). *Let , be nonempty subsets of metric space . The Hausdorff distance between and is defined by
where with . Let be a sequence of nonempty subsets of . We say that converges to in the sense of Hausdorff metric if . It is easy to see that if and only if for all selection . For more details on this topic, we refer the readers to [32].

#### 3. Bilevel Vector Equilibrium Problems with Equilibrium Constraints (BVEPEC)

In this section, we give some criteria and characterizations for LP well-posedness of (BVEPEC) using noncompactness. Now, we need the following notions of measures of noncompactness.

*Definition 7. *Let be a nonempty subset of a metric space . (i)The Kuratowski measure of is
(ii)The Hausdorff measure of is
(iii)The Istrǎtescu measure of is

The following inequalities are obtained in [33]: The measures , , and share many common properties and we will use in the sequel to denote that either one of them is a regular measure [34, 35]; that is, it enjoys the following properties: (i) if and only if the set is unbounded;(ii);(iii)if , then is a totally bounded set;(iv)if is a complete space and if is a sequence of closed subset of such that for each and , where is the Hausdorff metric;(v)if , then .

As above, denotes the solution set of . For positive , the -solution set of is defined by For positive and , the corresponding approximate solution sets for (BVEPEC) are defined, respectively, by

In terms of a measure of noncompactness we have the following result.

Theorem 8. *Let and be complete and . Assume that *(i)*in , is closed and is lsc;*(ii)* is upper -level closed in ;*(iii)* is upper 0-level closed in , for all ;*(iv)*the mapping defined by is closed.**Then, (BVEPEC) is type I LP well posed if and only if
*

*Proof. *By the relationship (14), the proof is similar for the three mentioned measures of noncompactness. We discuss only the case , the Kuratowski measure. Assume that (BVEPEC) is type I LP well posed. The solution set of (BVEPEC) clearly the relation . Hence,
Let be in . Since is an approximating sequence, it has a subsequence converging to some points of . Therefore, is compact.

Assume that with , . Setting , it obvious that and . Since is compact and , then we get
Next, we show that as . By contradiction, suppose the existence of , and such that , . This contradicts the type I LP well-posedness. So, as . It follows that (17) holds.

Conversely, first, we show that for all , is closed. Assume that as . Let with . Then, for all and ,
As is closed at , we have . From (20), we obtain that . By the upper -level closedness of in first argument and assumption (iv), one obtain
that is,
Next, we show by contrapositive that , . Suppose that there exist such that . Since is lsc at , there exist such that . By upper -level closedness of at , there is such that
That is a contradiction. Thus, we have
Therefore, and so is closed.

Secondly, we show that
It is obvious that . Now, suppose that and . Thus, we have
By (27) and (28) and closedness of , we obtain
That is, . Hence, (25) holds.

We know that as . Then, by properties of , we see that is compact and as . Let be a type I LP approximating solution sequence for (BVEPEC). There is such that, for all and ,
Therefore, . It follows that, from as ,
By the compactness of , there is a subsequence of convergent to some points of . Hence, (BVEPEC) is type I LP well posed. This completes the proof.

Similar to Theorem 8, we can prove that the following results.

Theorem 9. *Let and be complete and . Assume that *(i)*in , is closed and is lsc;*(ii)* is upper 0-level closed in ;*(iii)* is upper 0-level closed in , for all ;*(iv)*the mapping defined by is closed;*(v)*the set-valued mapping is closed;*(vi)*for any , and and for any , and .**
Then, (BVEPEC) is type II LP well posed if and only if as .*

#### 4. Optimization Problem with Equilibrium Constraints (OPEC)

In this section, we will present the criteria and characterization for four types of (BVEPEC) and introduce a gap function for (BVEPEC) using the nonlinear scalarization function and then we investigate the equivalent relations between the LP well-posedness for bilevel vector optimization problem with equilibrium constraints (BVOPEC) and the LP well-posedness for vector equilibrium problem with equilibrium constraints (BVEPEC). Now, consider the following optimization problem with equilibrium constraints.

Let be the solution map of (VQEP). Let , where . The bilevel vector optimization problem with equilibrium constraints is as follows: Note that is a parameter of the vector quasi-equilibrium problem defining the constraint, but it is a component of the decision variable of (BVEPEC) and (BVOPEC) and these problems are not parametric.

*Definition 10. *A sequence is called type I LP minimizing sequence for (BVOPEC) if and only if

*Definition 11. *A sequence is called type II LP minimizing sequence for (BVOPEC) if and only if (33) and (35) hold and

*Definition 12. *Problems (BVOPEC) is called type I (resp., type II) LP well posed if and only if (i)the solution set of (BVOPEC) is nonempty;(ii)for any type I (resp., type II), LP minimizing sequence of (BVOPEC) has a subsequence converging to a solution.

Now, we recall the definition of nonlinear scalarization function introduced by Chen et al. [36].

The nonlinear scalarization function is defined by

*Definition 13. *A mapping is called a gap function for (BVEPEC) if (i), ;(ii), if and only if .

We introduced the following gap function defined by

*Remark 14. *(i) By Definition 4, it is easy to see that is a gap function for (EPEC). Moreover, if , then .

(ii) By Definition 4, it is clear that if and only if minimizes over with .

Now, we prove the following lemma.

Lemma 15. *Let for any , , where is the topological boundary of and is upper -level closed on , for all . Then, the mapping defined by (38) is lower -level closed on .*

*Proof. *Suppose that satisfies and , . Follows from (38)
Then, , . By the upper closed -level of in first argument, we know that
That is,
Then, we have that is lower closed -level. This completes the proof.

Theorem 16. *Suppose that the assumptions of Lemma 15 are satisfied. Then the following results hold: *(i)*(BVEPEC) is the type I LP well-posedness if and only if (BVOPEC) is the type I LP well-posedness with the function defined by (38).*(ii)*(BVEPEC) is the type II LP well-posedness if and only if (BVOPEC) is the type II LP well-posedness with the function defined by (38).*

*Proof. *(i) We know that is a gap function of (BVEPEC), and if and only if with . Assume that is any type I LP approximating solution sequence for (BVEPEC). Then, there exist with such that
It follows from (43) that
Then, we obtain
Hence,
Therefore, is a type I LP minimizing sequence for (BVOPEC).

Conversely, assume that is any type I LP minimizing sequence for (BVOPEC). Then, , , and , . Then, there exist with satisfying ;
Then, we get or, equivalently,
Hence, is a type I LP approximating solution sequence for (BVEPEC). It follows that (BVEPEC) is the type I LP well-posedness if and only if (BVOPEC) is the type I LP well-posedness with the function .

The proof of (ii) is similar to (i) and we are omitted. This completes the proof.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to thank The Commission on Higher Education for financial support. Moreover, Kamonrat Sombut is also supported by The Strategic Scholarships Fellowships Frontier Research Networks under Grant no. CHE-Ph.D-THA-SUP/86/2550, Thailand.

#### References

- A. N. Tikhonov, “On the stability of the functional optimization problem,”
*USSR Computational Mathematics and Mathematical Physics*, vol. 6, no. 4, pp. 28–33, 1966. View at: Google Scholar - E. S. Levitin and B. T. Polyak, “Convergence of minimizing sequences in conditional extremum problems,”
*Soviet Mathematics Doklady*, vol. 7, pp. 764–767, 1966. View at: Google Scholar - Y.-P. Fang and R. Hu, “Parametric well-posedness for variational inequalities defined by bifunctions,”
*Computers & Mathematics with Applications*, vol. 53, no. 8, pp. 1306–1316, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. B. Lignola and J. Morgan, “Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution,”
*Journal of Global Optimization*, vol. 16, no. 1, pp. 57–67, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. B. Lignola, “Well-posedness and $L$-well-posedness for quasivariational inequalities,”
*Journal of Optimization Theory and Applications*, vol. 128, no. 1, pp. 119–138, 2006. View at: Publisher Site | Google Scholar | MathSciNet - R. Lucchetti,
*Recent Developments in Well-Posed Variational Problems*, Kluwer Academic, Dordrecht, The Netherlands, 1995. - R. Lucchetti and F. Patrone, “A characterization of Tyhonov well-posedness for minimum problems, with applications to variational inequalities,”
*Numerical Functional Analysis and Optimization*, vol. 3, no. 4, pp. 461–476, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. Lucchetti and F. Patrone, “Hadamard and Tyhonov well-posedness of a certain class of convex functions,”
*Journal of Mathematical Analysis and Applications*, vol. 88, no. 1, pp. 204–215, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Margiocco, F. Patrone, and L. Pusillo Chicco, “A new approach to Tikhonov well-posedness for Nash equilibria,”
*Optimization*, vol. 40, no. 4, pp. 385–400, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Margiocco, F. Patrone, and L. Pusillo Chicco, “Metric characterizations of Tikhonov well-posedness in value,”
*Journal of Optimization Theory and Applications*, vol. 100, no. 2, pp. 377–387, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Margiocco, F. Patrone, and L. Pusillo, “On the Tikhonov well-posedness of concave games and Cournot oligopoly games,”
*Journal of Optimization Theory and Applications*, vol. 112, no. 2, pp. 361–379, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Morgan, “Approximations and well-posedness in multicriteria games,”
*Annals of Operations Research*, vol. 137, pp. 257–268, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. S. Konsulova and J. P. Revalski, “Constrained convex optimization problems—well-posedness and stability,”
*Numerical Functional Analysis and Optimization*, vol. 15, no. 7-8, pp. 889–907, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Beer and R. Lucchetti, “The epi-distance topology: continuity and stability results with applications to convex optimization problems,”
*Mathematics of Operations Research*, vol. 17, no. 3, pp. 715–726, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. X. Huang and X. Q. Yang, “Levitin-Polyak well-posedness of constrained vector optimization problems,”
*Journal of Global Optimization*, vol. 37, no. 2, pp. 287–304, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. J. Li and M. H. Li, “Levitin-Polyak well-posedness of vector equilibrium problems,”
*Mathematical Methods of Operations Research*, vol. 69, no. 1, pp. 125–140, 2009. View at: Publisher Site | Google Scholar - N.-J. Huang, X.-J. Long, and C.-W. Zhao, “Well-posedness for vector quasi-equilibrium problems with applications,”
*Journal of Industrial and Management Optimization*, vol. 5, no. 2, pp. 341–349, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. H. Li, S. J. Li, and W. Y. Zhang, “Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems,”
*Journal of Industrial and Management Optimization*, vol. 5, no. 4, pp. 683–696, 2009. View at: Publisher Site | Google Scholar - J. V. Outrata, “A generalized mathematical program with equilibrium constraints,”
*SIAM Journal on Control and Optimization*, vol. 38, no. 5, pp. 1623–1638, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. S. Mordukhovich, “Characterizations of linear suboptimality for mathematical programs with equilibrium constraints,”
*Mathematical Programming*, vol. 120, no. 1, pp. 261–283, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. B. Lignola and J. Morgan, “$\alpha $-well-posedness for Nash equilibria and for optimization problems with Nash equilibrium constraints,”
*Journal of Global Optimization*, vol. 36, no. 3, pp. 439–459, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Q. Bao, P. Gupta, and B. S. Mordukhovich, “Necessary conditions in multiobjective optimization with equilibrium constraints,”
*Journal of Optimization Theory and Applications*, vol. 135, no. 2, pp. 179–203, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Q. Anh, P. Q. Khanh, and D. T. M. Van, “Well-posedness under relaxed semicontinuity for bilevel equilibrium and optimization problems with equilibrium constraints,”
*Journal of Optimization Theory and Applications*, vol. 153, no. 1, pp. 42–59, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q. H. Ansari, “Vector equilibrium problems and vector variational inequalities,” in
*Vector Variational Inequalities and Vector Equilibria*, F. Giannessi, Ed., Mathematical Theories, p. 116, Kluwer, Dordrecht, The Netherlands, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Giannessi, Ed.,
*Vector Variational Inequalities and Vector Equilibria*, vol. 38 of*Nonconvex Optimization and its Applications*, Kluwer Academic, Dordrecht, The Netherlands, 2000. View at: Publisher Site | MathSciNet - G. Y. Chen, X. Q. Yang, and H. Yu, “A nonlinear scalarization function and generalized quasi-vector equilibrium problems,”
*Journal of Global Optimization*, vol. 32, no. 4, pp. 451–466, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. J. Li, K. L. Teo, and X. Q. Yang, “Generalized vector quasi-equilibrium problems,”
*Mathematical Methods of Operations Research*, vol. 61, no. 3, pp. 385–397, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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,”
*Journal of Global Optimization*, vol. 34, no. 3, pp. 427–440, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q. H. Ansari and F. Flores-Bazán, “Generalized vector quasi-equilibrium problems with applications,”
*Journal of Mathematical Analysis and Applications*, vol. 277, no. 1, pp. 246–256, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Bianchi, N. Hadjisavvas, and S. Schaible, “Vector equilibrium problems with generalized monotone bifunctions,”
*Journal of Optimization Theory and Applications*, vol. 92, no. 3, pp. 527–542, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. Song, “On generalized vector equilibrium problems,”
*Applied Mathematics Letters*, vol. 12, pp. 53–56, 2002. View at: Google Scholar - C. Kuratowski,
*Topology. Panstwowe Wydawnicto Naukowa*, vol. 1, Warszawa, Poland, 1958. - J. Daneš, “On the Istrătescu measure of noncompactness,”
*Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie*, vol. 16, no. 4, pp. 403–406, 1972. View at: Google Scholar | MathSciNet - J. Banaś and K. Goebel,
*Measures of Noncompactness in Banach Spaces*, vol. 60 of*Lecture Notes in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 1980. View at: MathSciNet - V. Rakočević, “Measures of noncompactness and some applications,”
*Filomat*, no. 12, pp. 87–120, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet - G.-Y. Chen, X. Huang, and X. Yang,
*Vector Optimization, Set-Valued and Variational Analysis*, vol. 541 of*Lecture Notes in Economics and Mathematical Systems*, Springer, Berlin, Germany, 2005. View at: MathSciNet

#### Copyright

Copyright © 2014 Phan Quoc Khanh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.