Piecewise Linear Vector Optimization Problems on Locally Convex Hausdorff Topological Vector Spaces
- 158 Downloads
Piecewise linear vector optimization problems in the locally convex Hausdorff topological vector space setting are considered in this paper. The efficient solution set of these problems are shown to be the unions of finitely many semi-closed generalized polyhedral convex sets. If, in addition, the problem is convex, then the efficient solution set and the weakly efficient solution set are the unions of finitely many generalized polyhedral convex sets and they are connected by line segments. Our results develop the preceding ones of Zheng and Yang (Sci. China Ser. A. 51, 1243–1256 2008), and Yang and Yen (J. Optim. Theory Appl. 147, 113–124 2010), which were established in the normed space setting.
KeywordsLocally convex Hausdorff topological vector space Generalized polyhedral convex set Piecewise linear vector optimization problem Semi-closed generalized polyhedral convex set Connectedness by line segments
Mathematics Subject Classification (2010)90C29 90C30 90C48
The author would like to thank Professor Nguyen Dong Yen for his guidance and the anonymous referees for valuable suggestions.
This research was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.37.
- 1.Arrow, K.J., Barankin, E.W., Blackwell, D.: Admissible points of convex sets. In: Contributions to The Theory of Games, vol. 2. Annals of Mathematics Studies 28, pp 87–91. Princeton University Press, Princeton (1953)Google Scholar
- 2.Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer (2000)Google Scholar
- 3.Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization. Set-Valued and Variational Analysis Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)Google Scholar
- 11.Luan, N.N.: Efficient solutions in generalized linear vector optimization. Preprint (arXiv:https://arxiv.org/abs/1705.06875); submitted
- 12.Luan, N.N., Yao, J.-C., Yen, N.D.: On some generalized polyhedral convex constructions. Numerical Functional Analysis and Optimization, https://doi.org/10.1080/01630563.2017.1387863
- 13.Luan, N.N., Yen, N.D.: A representation of generalized convex polyhedra and applications. Preprint (arXiv:https://arxiv.org/abs/1705.06874); submitted
- 14.Luc, D.T.: Theory of Vector Optimization Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989)Google Scholar
- 15.Luc, D.T.: Multiobjective Linear Programming. An Introduction. Springer Cham (2016)Google Scholar
- 16.Luc, D.T., Raţiu, A.: Vector optimization: basic concepts and solution methods. In: Al-Mezel, S. A. R., Al-Solamy, F. R. M., Ansari, Q. (eds.) Fixed Point Theory, Variational Analysis, and Optimization, pp 249–305. CRC Press, Boca Raton (2014)Google Scholar
- 20.Rudin, W.: Functional Analysis, 2nd edn. McGraw Hill (1991)Google Scholar
- 23.Weyl, H: The elementary theory of convex polyhedra. In: Contributions to the Theory of Games, Annals of Mathematics Studies, vol. 24, pp 3–18. Princeton University Press, Princeton (1950)Google Scholar