International Journal of Nonlinear Analysis and Applications

Finding properly efficient solutions of nonconvex multiobjective optimization problems with a minimum bound for trade-offs

Document Type : Research Paper

Authors

1 Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran

2 Faculty of Mathematical Sciences, shahrood University of Technology, Shahrood, Iran

Abstract

In the presented paper, we investigate efficient solutions to optimization problems with multiple criteria and bounded trade-offs. A nonlinear optimization problem to find the relationships between the upper bound for trade-offs and objective functions is presented. Due to this problem, we determine some properly efficient points that are closer to the ideal point. To this end,  we apply the extended form of the generalized Tchebycheff norm. Note that all the presented results work for general problems and no convexity assumption is needed.

Keywords

[1] F. Akbari, M. Ghaznavi, and E. Khorram, A revised Pascoletti–Serafini scalarization method for multiobjective optimization problems, J. Optim. Theory Appl. 178 (2018), no. 2, 560–590.
[2] H.P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones, J. Math. Anal. Appl. 71 (1979), 232–241.
[3] J.M. Borwein, Proper efficient points for maximization with respect to cones, SIAM J. Control Optim. 15 (1977), 57–63.
[4] E.U. Choo and D.R. Atkins, Proper efficiency in nonconvex multicriteria programming, Math. Oper. Res. 8 (1983), 467–470.
[5] S. Dempe, Foundations of Bilevel Programming, Springer, Berlin, 2002.  [6] M. Ehrgott, Multicriteria Optimization, Springer, Berlin, 2005.
[7] P. Eskelinen and K. Miettinen, Trade-off analysis approach for interactive nonlinear multiobjective optimization, OR Spectrum 34 (2012), 803–816.
[8] A.M. Geoffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl. 22 (1968), 618–630. [9] M. Henig, Proper efficiency with respect to cones, J. Optim. Theory Appl. 36 (1982), no. 3, 387–407.
[10] N. Hoseinpoor and M. Ghaznavi, The modified objective-constraint scalarization approach for multiobjective optimization problems, Hacet. J. Math. Stat. 51 (2022), no. 5, 1403–1418.
[11] B. Hozzar, G.H. Tohidi, and B. Daneshian, Two methods for determining properly efficient solutions with a minimum upper bound for trade-offs, Filomat 33 (2019), no. 6, 1551–1559.
[12] V. Jog, I. Kaliszewski and W. Michalowski, Using trade-off information in attributes’ investing, IIASA Interm Report, No. IR-98–19, 1998.
[13] M. Karimi and B. Karimi, Linear and conic scalarizations for obtaining properly efficient solutions in multiobjective optimization, Math. Sci. 11 (2017), 319–325.
[14] K. Khaledian, E. Khorram, and M. Soleimani-damaneh, Strongly proper efficient solutions: efficient solutions with bounded trade-offs, J. Optim. Theory Appl. 168 (2016), 864–883.
[15] K. Khaledian and M. Soleimani-damaneh, On efficient solutions with trade-offs bounded by given values, Numer. Funct. Anal. Optim. 36 (2015), no. 11, 1431–1447.
[16] H. Kuhn and A. Tucker, Nonlinear programming, Proc. Second Berkeley Symp. Math. Statist. Probab. J. Neyman, 1951, pp. 481–492.
[17] K. Miettinen, Nonlinear Multiobjective Optimization, International Series in Operations Research and Management Science, Vol. 12. Kluwer Academic Publishers, Dordrecht, 1999.
[18] L. Pourkarimi and M. Karimi, Characterization of substantially and quasi-substantially efficient solutions in multiobjective optimization problems, Turk. J. Math. 41 (2017), 293–304.
[19] P.K. Shukla, J. Dutta, K. Deb, and P. Kesarwani, On a practical notion of Geoffrion proper optimality in multicriteria optimization, Optimization 69 (2020), no. 7-8, 1513–1539.
[20] M. Soleimani-damaneh and M. Zamani, On compromise solutions in multiple objective programming, RAIROOper. Res. 52 (2018), 383–390.
[21] R.E. Steuer, Multiple criteria optimization: theory, computation, and application, Wiley, New York, 1986.
[22] G.G. Tejani, N. Pholdee, S. Bureerat, D. Prayogo, and A.H. Gandomi, Structural optimization using multiobjective modified adaptive symbiotic organisms search, Expert Syst. Appl. 125 (2019), 425–441.
[23] P. Wang, J. Huang, Z. Cui, L. Xie, and J. Chen, A Gaussian error correction multiobjective positioning model with NSGA-II, Concurr. Comput. Pract. Exp. 32 (2020), no. 5.
[24] S.C. Wang and T.C. Chen, Multi-objective competitive location problem with distance-based attractiveness and its best non-dominated solution, Appl. Math. Model. 47 (2017), 785-795.
[25] SC. Wang, H.C.W. Hsiao, C.C. Lin and H.H. Chin, Multi-objective wireless sensor network deployment problem with cooperative distance-based sensing coverage, Mobile Netw. Appl. 27 (2022), 3–14.
[26] Y.M. Xia, X.M. Yang, and K.Q. Zhao, A combined scalarization method for multi-objective optimization problems, J. Ind. Manag. Optim. 17 (2021), no. 5, 2669–2683.
[27] M. Zamani and M. Soleimani-Damaneh, Proper efficiency, scalarization and transformation in multi-objective optimization: Unified approaches, Optimization 71 (2022), no. 3, 753–774.
International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 5
May 2024
Pages 239-245
  • Receive Date: 11 January 2023
  • Accept Date: 14 June 2023