A new approach to zero duality gap of vector optimization problems using characterizing sets
Main Article Content
Abstract
In this paper we propose results on zero duality gap in vector optimization problems posed in a real locally convex Hausdorff topological vector space with a vector-valued objective function to be minimized under a set and a convex cone constraint. These results are then applied to linear programming.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Keywords
Characterizing set, vector optimization problems, zero dualiy gap
References
Anderson, E.J. (1983). A review of duality theory for linear programming over topological vector spaces. J. Math. Anal. Appl., 97(2), 380-392.
Andreas, L. (2011). Vector optimization with infimum and supremum. Berlin: Springer- Verlag.
Bot, R.I. (2010). Conjugate duality in convex optimization. Berlin: Springer.
Bot, R.I., Grad, S.M., & Wanka, G. (2009): Duality in Vector Optimization. Berlin: Springer-Verlag.
Cánovas, M. J., Dinh, N., Long, D. H., & Parra, J. (2021). A new approach to strong duality for composite vector optimization problems. Optimization. Optimization. [10.1080/02331934.2020.1745796].
Elvira, H., Andreas, L., Luis, R., & Tammer, C. (2013). Lagrange duality, stability and subdifferentials in vector optimization.
Optimization, 62(3), 415-428.
Jeyakumar, V., & Volkowicz, H. (1990). Zero duality gap in infinite-dimensional programming. J. Optim. Theory Appl., 67(1), 88-108.
Khan, A., Tammer, C., & Zalinescu, C. (2005). Set-valued optimization: An introduction with applications. Heidelberg: Springer.
Nguyen, D., Long, D. H., Mo, T. H., & Yao, J. C. (2020). Approximate Farkas lemmas for vector systems with applications to convex vector optimization problems. J. Non. Con. Anal., 21(5), 1225-1246.
Nguyen, D., & Dang, H. L. (2018). Complete characterizations of robust strong duality for robust vector optimization problems. Vietnam J. Math., 46(2), 293-328. https://doi.org/10.1007/s10013-018-0283-1.
Nguyen, D., Goberna, M. A., López, M. A., & Tran, H. M. (2017). Farkas-type results for vector-valued functions with applications. Journal of Optimization Theory and Applications, 173, 357-390. https://doi.org/10.1007/s10957-016-1055-2.
Nguyen, D., Goberna, M.A., Dang, H. L., & Lopez, M.A. (2019). New Farkas-type results for vector-valued functions: A non- abstract approach. J. Optim. Theory Appl., 82(1), 4-29. https://doi.org/10.1007/s10957-018-1352-z.
Nguyen, T. V., Kim, D.S., Nguyen, N. T., & Nguyen, D. Y. (2016). Duality gap function in infinite dimensional linear programming. J. Math. Anal. Appl., 437(1), 1-15. https://doi.org/10.1016/j.jmaa.2015.12.043
Pham, D. K., Tran, H. M., & Tran, T. T. T. (2019). Necessary and sufficient conditions for qualitative properties of infinite dimensional linear programming problems. Numer. Funct. Anal. Optim., 40(8), 924-943. https://doi.org/10.1080/01630563.2019.1566244.
Rudin, W. (1991). Functional analysis (2nd Edition). New York: McGraw-Hill.
Tanino, T. (1992). Conjugate duality in vector optimization. J. Math. Anal. Appl., 167(1), 84-97.
Wen, S. (1998). Duality in set-valued optimization. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk.
Zalinescu, C. (2002). Convex analysis in general vector spaces. Singapore: World Scientificc Publishing.
Andreas, L. (2011). Vector optimization with infimum and supremum. Berlin: Springer- Verlag.
Bot, R.I. (2010). Conjugate duality in convex optimization. Berlin: Springer.
Bot, R.I., Grad, S.M., & Wanka, G. (2009): Duality in Vector Optimization. Berlin: Springer-Verlag.
Cánovas, M. J., Dinh, N., Long, D. H., & Parra, J. (2021). A new approach to strong duality for composite vector optimization problems. Optimization. Optimization. [10.1080/02331934.2020.1745796].
Elvira, H., Andreas, L., Luis, R., & Tammer, C. (2013). Lagrange duality, stability and subdifferentials in vector optimization.
Optimization, 62(3), 415-428.
Jeyakumar, V., & Volkowicz, H. (1990). Zero duality gap in infinite-dimensional programming. J. Optim. Theory Appl., 67(1), 88-108.
Khan, A., Tammer, C., & Zalinescu, C. (2005). Set-valued optimization: An introduction with applications. Heidelberg: Springer.
Nguyen, D., Long, D. H., Mo, T. H., & Yao, J. C. (2020). Approximate Farkas lemmas for vector systems with applications to convex vector optimization problems. J. Non. Con. Anal., 21(5), 1225-1246.
Nguyen, D., & Dang, H. L. (2018). Complete characterizations of robust strong duality for robust vector optimization problems. Vietnam J. Math., 46(2), 293-328. https://doi.org/10.1007/s10013-018-0283-1.
Nguyen, D., Goberna, M. A., López, M. A., & Tran, H. M. (2017). Farkas-type results for vector-valued functions with applications. Journal of Optimization Theory and Applications, 173, 357-390. https://doi.org/10.1007/s10957-016-1055-2.
Nguyen, D., Goberna, M.A., Dang, H. L., & Lopez, M.A. (2019). New Farkas-type results for vector-valued functions: A non- abstract approach. J. Optim. Theory Appl., 82(1), 4-29. https://doi.org/10.1007/s10957-018-1352-z.
Nguyen, T. V., Kim, D.S., Nguyen, N. T., & Nguyen, D. Y. (2016). Duality gap function in infinite dimensional linear programming. J. Math. Anal. Appl., 437(1), 1-15. https://doi.org/10.1016/j.jmaa.2015.12.043
Pham, D. K., Tran, H. M., & Tran, T. T. T. (2019). Necessary and sufficient conditions for qualitative properties of infinite dimensional linear programming problems. Numer. Funct. Anal. Optim., 40(8), 924-943. https://doi.org/10.1080/01630563.2019.1566244.
Rudin, W. (1991). Functional analysis (2nd Edition). New York: McGraw-Hill.
Tanino, T. (1992). Conjugate duality in vector optimization. J. Math. Anal. Appl., 167(1), 84-97.
Wen, S. (1998). Duality in set-valued optimization. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk.
Zalinescu, C. (2002). Convex analysis in general vector spaces. Singapore: World Scientificc Publishing.
Most read articles by the same author(s)
- Hai Long Dang, Hong Mo Tran, A unified approach to zero duality gap for convex optimization problems , Dong Thap University Journal of Science: Vol. 11 No. 5 (2022): Natural Sciences Issue (English)