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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
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.
Bianchi, M., Hadjisavvas, N., & Schaible, S. (1997). Vector equilibrium problems with generalized monotone bifunctions. Journal of Optimization Theory and Applications, 92(3), 527–542.
Bigi, G., Capătă, A., & Kassay, G. (2012). Existence results for strong vector equilibrium problems and their applications. Optimization, 61(5), 567–583.
Blum, E., & Oettli, W. (1994). From optimization and variational inequalities to equilibrium problems. The Mathematics Student, 63(1–4), 123–145.
Breckner, W. W., & Kassay, G. (1997). A systematization of convexity concepts for sets and functions. Journal of Convex Analysis, 4(1), 109–127.
Capătă, A. (2011). Existence results for proper efficient solutions of vector equilibrium problems and applications. Journal of Global Optimization, 51(4), 657–675.
Debreu, G. (1959). Theory of value. New York: Wiley.
Geoffrion, A. M. (1968). Proper efficiency and the theory of vector maximization. Journal of Mathematical Analysis and Applications, 22(3), 618–630.
Göpfert, A., Riahi, H., Tammer, Chr, & Zălinescu, C. (2003). Variational methods in partially ordered spaces. New York: Springer.
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.
Henig, M. I. (1982). Proper efficiency with respect to cones. Journal of Optimization Theory and Applications, 36(3), 387–407.
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.
Iusem, A. N., & Sosa, W. (2003). New existence results for equilibrium problems. Nonlinear Analysis: Theory, Methods & Applications, 52(2), 621–635.
Jeyakumar, V. (1985). Convexlike alternative theorems and mathematical programming. Optimization, 16(5), 643–652.
La Torre, D., & Popovici, N. (2010). Arcwise cone-quasiconvex multicriteria optimization. Operations Research Letters, 38(2), 143–146.
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.
Luc, D. T. (1989). Theory of vector optimization. Lecture notes in economics and mathematical systems, 319. Berlin: Springer.
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.
Oettli, W. (1997). A remark on vector-valued equilibria and generalized monotonicity. Acta Mathematica Vietnamica, 22(1), 213–221.
Podinovskiĭ, V. V., & Nogin, V. D. (1982). Pareto optimal solutions of multicriteria optimization problems (in Russian). Moscow: Nauka.
Popovici, N. (2005). Pareto reducible multicriteria optimization problems. Optimization, 54(3), 253–263.
Popovici, N., & Rocca, M. (2012). Decomposition of generalized vector variational inequalities. Nonlinear Analysis: Theory, Methods & Applications, 75(3), 1516–1523.
Popovici, N., & Rocca, M. (2013). Scalarization and decomposition of vector variational inequalities governed by bifunctions. Optimization, 62(6), 735–742.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Popovici, N. A decomposition approach to vector equilibrium problems. Ann Oper Res 251, 105–115 (2017). https://doi.org/10.1007/s10479-015-1861-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-015-1861-1