# A decomposition approach to vector equilibrium problems

- 144 Downloads

## Abstract

A vector equilibrium problem is generally defined by a bifunction which takes values in a partially ordered vector space. When this space is endowed with a componentwise ordering, the vector equilibrium problem can be decomposed into a family of equilibrium subproblems, each of them being governed by a bifunction obtained from the initial one by selecting some of its scalar components. Similarly to multi-criteria optimization, three types of solutions can be defined for these equilibrium subproblems, namely, weak, strong and proper solutions. The aim of this paper is to show that, under appropriate convexity assumptions, the set of all weak solutions of a vector equilibrium problem can be recovered as the union of the sets of proper solutions of its subproblems.

## Keywords

Vector equilibrium problem Multi-criteria optimization problem Generalized convexity Scalarization Decomposition## Notes

### Acknowledgments

This research was partially supported by CNCS–UEFISCDI, Project PN-II-ID-PCE-2011-3-0024. The author is grateful to the referees for valuable comments.

## References

- Ansari, Q. H. (2000). Vector equilibrium problems and vector variational inequalities. In F. Giannessi (Ed.),
*Vector variational inequalities and vector equilibria. Mathematical theories*(pp. 1–16). Dordrecht, Boston: Kluwer.CrossRefGoogle Scholar - Bianchi, M., Hadjisavvas, N., & Schaible, S. (1997). Vector equilibrium problems with generalized monotone bifunctions.
*Journal of Optimization Theory and Applications*,*92*(3), 527–542.CrossRefGoogle Scholar - Bigi, G., Capătă, A., & Kassay, G. (2012). Existence results for strong vector equilibrium problems and their applications.
*Optimization*,*61*(5), 567–583.CrossRefGoogle Scholar - Blum, E., & Oettli, W. (1994). From optimization and variational inequalities to equilibrium problems.
*The Mathematics Student*,*63*(1–4), 123–145.Google Scholar - Breckner, W. W., & Kassay, G. (1997). A systematization of convexity concepts for sets and functions.
*Journal of Convex Analysis*,*4*(1), 109–127.Google Scholar - Capătă, A. (2011). Existence results for proper efficient solutions of vector equilibrium problems and applications.
*Journal of Global Optimization*,*51*(4), 657–675.CrossRefGoogle Scholar - Debreu, G. (1959).
*Theory of value*. New York: Wiley.Google Scholar - Geoffrion, A. M. (1968). Proper efficiency and the theory of vector maximization.
*Journal of Mathematical Analysis and Applications*,*22*(3), 618–630.CrossRefGoogle Scholar - Göpfert, A., Riahi, H., Tammer, Chr, & Zălinescu, C. (2003).
*Variational methods in partially ordered spaces*. New York: Springer.Google Scholar - Guerraggio, A., Molho, E., & Zaffaroni, A. (1994). On the notion of proper efficiency in vector optimization.
*Journal of Optimization Theory and Applications*,*82*(1), 1–21.CrossRefGoogle Scholar - Henig, M. I. (1982). Proper efficiency with respect to cones.
*Journal of Optimization Theory and Applications*,*36*(3), 387–407.CrossRefGoogle Scholar - Hurwicz, L. (1958). Programming in linear spaces. In K. J. Arrow, L. Hurwicz, & H. Uzawa (Eds.),
*Studies in linear and non-linear programming*(pp. 38–102). Stanford: Stanford University Press.Google Scholar - Iusem, A. N., & Sosa, W. (2003). New existence results for equilibrium problems.
*Nonlinear Analysis: Theory, Methods & Applications*,*52*(2), 621–635.CrossRefGoogle Scholar - Jeyakumar, V. (1985). Convexlike alternative theorems and mathematical programming.
*Optimization*,*16*(5), 643–652.CrossRefGoogle Scholar - La Torre, D., & Popovici, N. (2010). Arcwise cone-quasiconvex multicriteria optimization.
*Operations Research Letters*,*38*(2), 143–146.CrossRefGoogle Scholar - Lowe, T. J., Thisse, J.-F., Ward, J. E., & Wendell, R. E. (1984). On efficient solutions to multiple objective mathematical programs.
*Management Science*,*30*(11), 1346–1349.CrossRefGoogle Scholar - Luc, D. T. (1989).
*Theory of vector optimization*. Lecture notes in economics and mathematical systems, 319. Berlin: Springer.Google Scholar - Muu, Lê D., & Oettli, W. (1992). Convergence of an adaptive penalty scheme for finding constrained equilibria.
*Nonlinear Analysis: Theory, Methods & Applications*,*18*(12), 1159–1166.CrossRefGoogle Scholar - Oettli, W. (1997). A remark on vector-valued equilibria and generalized monotonicity.
*Acta Mathematica Vietnamica*,*22*(1), 213–221.Google Scholar - Podinovskiĭ, V. V., & Nogin, V. D. (1982).
*Pareto optimal solutions of multicriteria optimization problems (in Russian)*. Moscow: Nauka.Google Scholar - Popovici, N. (2005). Pareto reducible multicriteria optimization problems.
*Optimization*,*54*(3), 253–263.CrossRefGoogle Scholar - Popovici, N., & Rocca, M. (2012). Decomposition of generalized vector variational inequalities.
*Nonlinear Analysis: Theory, Methods & Applications*,*75*(3), 1516–1523.CrossRefGoogle Scholar - Popovici, N., & Rocca, M. (2013). Scalarization and decomposition of vector variational inequalities governed by bifunctions.
*Optimization*,*62*(6), 735–742.CrossRefGoogle Scholar