Abstract
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.
Similar content being viewed by others
Notes
Some authors use the term “K-convex function” instead of K-function.
References
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)
Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer (2000)
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)
Fang, Y.P., Huang, N.J., Yang, X.Q.: Local smooth representations of parametric semiclosed polyhedra with applications to sensitivity in piecewise linear programs. J. Optim. Theory Appl. 155, 810–839 (2012)
Fang, Y.P., Meng, K.W., Yang, X.Q.: Piecewise linear multi-criteria programs: the continuous case and its discontinuous generalization. Oper. Res. 60, 398–409 (2012)
Fang, Y.P., Meng, K.W., Yang, X.Q.: On minimal generators for semiclosed polyhedra. Optimization 64, 761–770 (2015)
Giannessi, F.: Theorem of the alternative and optimality conditions. J. Optim. Theory Appl. 42, 331–365 (1984)
Giannessi, F.: Theorems of the alternative for multifunctions with applications to optimization: General results. J. Optim. Theory Appl. 55, 233–256 (1987)
Klee, V.: Some characterizations of convex polyhedra. Acta. Math. 102, 79–107 (1959)
Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Dover Publications, New York (1975)
Luan, N.N.: Efficient solutions in generalized linear vector optimization. Preprint (arXiv:https://arxiv.org/abs/1705.06875); submitted
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
Luan, N.N., Yen, N.D.: A representation of generalized convex polyhedra and applications. Preprint (arXiv:https://arxiv.org/abs/1705.06874); submitted
Luc, D.T.: Theory of Vector Optimization Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989)
Luc, D.T.: Multiobjective Linear Programming. An Introduction. Springer Cham (2016)
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)
Luenberger, D.: Optimization by Vector Space Methods. Wiley, New York (1969)
Minkowski, H: Geometrie der Zahlen. Teubner, Leipzig–Berlin (1910)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Rudin, W.: Functional Analysis, 2nd edn. McGraw Hill (1991)
Schaefer, H.H.: Topological Vector Spaces Graduate Texts in Mathematics, vol. 3. Springer, New York-Berlin (1971)
Weyl, H.: Elementare Theorie der konvexen Polyeder. Commentarii Math. Helvetici 7, 290–306 (1935)
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)
Yang, X.Q., Yen, N.D.: Structure and weak sharp minimum of the Pareto solution set for piecewise linear multiobjective optimization. J. Optim. Theory Appl. 147, 113–124 (2010)
Zheng, X.Y., Ng, K.: Metric subregularity of piecewise linear multifunctions and applications to piecewise linear multiobjective optimization. SIAM J. Optim. 24, 154–174 (2014)
Zheng, X.Y., Yang, X.Q.: The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces. Sci. China Ser. A. 51, 1243–1256 (2008)
Acknowledgements
The author would like to thank Professor Nguyen Dong Yen for his guidance and the anonymous referees for valuable suggestions.
Funding
This research was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.37.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Luan, N. Piecewise Linear Vector Optimization Problems on Locally Convex Hausdorff Topological Vector Spaces. Acta Math Vietnam 43, 289–308 (2018). https://doi.org/10.1007/s40306-017-0239-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40306-017-0239-7
Keywords
- Locally convex Hausdorff topological vector space
- Generalized polyhedral convex set
- Piecewise linear vector optimization problem
- Semi-closed generalized polyhedral convex set
- Connectedness by line segments