Piecewise Linear Vector Optimization Problems on Locally Convex Hausdorff Topological Vector Spaces

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.

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Notes

  1. 1.

    Some authors use the term “K-convex function” instead of K-function.

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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.

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Correspondence to Nguyen Ngoc Luan.

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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

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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

Mathematics Subject Classification (2010)

  • 90C29
  • 90C30
  • 90C48