Approximate proper solutions in vector optimization with variable ordering structure

Document Type : Research Article


Department of Mathematics, Faculty of Statistics, Mathematics, and Computer Science, Allameh Tabataba’i University , Tehran, Iran.


In this paper, we study approximate proper efficient (nondominated and minimal) solutions of vector optimization problems with variable ordering structures (VOSs). In vector optimization with VOS, the partial order-ing cone depends on the elements of the image set. Approximate proper efficient/nondominated/ minimal solutions are defined in different senses (Henig, Benson, and Borwein) for problems with VOSs from new stand-points. The relationships among the introduced notions are studied, and some scalarization approaches are developed to characterize these solutions. These scalarization results based on new functionals defined by elements from the dual cones are given. Moreover, some existing results are ad-dressed.


Main Subjects

[1] Adan, M. and Novo, V. Proper efficiency in vector optimization on real linear spaces, J. Optim. Theory Appl. 121 (2004), 515–540.
[2] Bao, T.Q., Mordukhovich, B.S. and Soubeyran, A. Variational analysis in psychological modeling, J. Optim. Theory Appl. 164 (2015), 290–315.
[3] Bazgan, C., Ruzika, S., Thielen, C. and Vanderpooten, D. The power of the weighted sum scalarization for approximating multiobjective opti-mization problems , Theory Comput. Syst. 66 (2022), 395–415.
[4] Benson, H.P. An improved definition of proper efficiency for vector max-imization with respect to cone, J. Math. Appl. 71 (1979), 232–241.
[5] Bergstresser, K., Charnes, A. and Yu, PL. Generalization of domination structures and nondominated solutions in multicriteria decision making, J. Optim. Theory Appl., 18 (1976), 3–13.
[6] Borwein, J.M. Proper efficient points for maximizations with respect to cones, SIAM J. Control Optim. 15 (1977), 57–63.
[7] Chen, G.Y. Existence of solutions for a vector variational inequality: an extension of the Hartmann Stampacchia theorem, J. Optim. Theory Appl. 74 (1992), 445–456.
[8] Chen, G.Y. and Craven, B.D. Existence and continuity of solutions for vector optimization, J. Optim. Theory Appl. 81(3) (1994), 459–468.
[9] Chen, G.Y., Huang, X. and Yang, X. Vector Optimization, Set-Valued and Variational Analysis, Springer, Berlin, 2005.
[10] Chen, G.Y. and Yang, X.Q. Characterizations of variable domination structures via nonlinear scalarization, J. Optim. Theory Appl. 112 (2002), 97–110.
[11] Dauer, J.P. and Saleh, O.A. A characterization of proper minimal points as solutions of sublinear optimizations problems, J. Math. Anal. Appl. 178 (1993), 227–246.
[12] Durea, M., Florea E.A. and Strugriu, R. Henig proper efficiency in vector optimization with variable ordering structure, J. Ind. Manag. Optim. 15(2) (2019), 791–815.
[13] Eichfelder, G. Optimal elements in vector optimization with a variable ordering structure, J. Optim. Theory Appl. 151 (2011), 217–240.
[14] Eichfelder, G. Vector optimization in medical engineering, in Pardalos PM, Rassias TM (eds.) Mathematics Without Boundaries Springer, New York, (2014), 181–215.
[15] Eichfelder, G. Variable Ordering Structures in Vector Optimization, Springer, Berlin, 2014.
[16] Eichfelder, G. and Gerlach, T. Characterization of properly optimal el-ements with variable ordering structures, Optimization, 65 (2016), 571–588.
[17] Eichfelder, G., Kasimbeyli, R. Properly optimal elements in vector op-timization with variable ordering structure, J. Global Optim. 60 (2014), 689–712.
[18] Engau, A. Variable preference modeling with ideal-symmetric convex cones, J. Global Optim. 42 (2008), 295–311.
[19] Foroutannia, D. and Mahmodinejad, A. The concept of B-efficient solu-tion in fair multiobjective optimization problems, Iran. J. Numer. Anal. Optim. 7(1) (2017), 47–63.
[20] Geoffrion, A. Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl. 22 (1968), 618–630.
[21] Ghaznavi-Ghosoni, B.A., Khorram E. and Soleimani-Damaneh, M. Scalarization for characterization of approximate strong/weak/proper efficiency in multi-objective optimization, Optimization, 62(6) (2013), 703–720.
[22] Guerraggio, A., Molho, E. and Zaffaroni, A. On the notion of proper efficiency in vector optimization, J. Optim. Theory Appl. 82 (1994), 1–21.
[23] Helbig, S. On a new concept for ε-efficiency, talk at Optimization Days 1992, Montreal, (1992).
[24] Helbig, S. and Pateva, D. On several concepts for ε-efficiency, OR Spec-trum, 16 (1994), 179–186.
[25] Henig, M.I. Proper efficiency with respect to cones, J. Optim. Theory Appl. 36 (1982), 387–407.
[26] Hoseinpour, N. and Ghaznavi, M. Identifying approximate proper effi-ciency in an infinite dimensional space, RARO Operations Research, 57 (2023) 697–714.
[27] Jahn, J. Vector optimization, Springer, Berlin, 2012.
[28] Kiyani, E. and Soleimani-damaneh, M. Approximate proper efficiency on real linear vector spaces, Pacific Journal of Optimization, 10 (2014), 715–734.
[29] Kuhn, H. and Tucker, A. Nonlinear programming, in Neyman, J. (ed.) Proceeding of the Second Berkeley Symposium on Mathematical Statis-tics and Probability, University of California Press, Berkeley, California, (1951) 481–492.
[30] Kutateladze, S.S. convex ε-programming, Soviet. Math. Dokl, 20 (1979), 391–393.
[31] Luc, D.T. and Soubeyran, A. Variable preference relations: existence of maximal elements, J. Math. Econ. 49 (2013), 251–262.
[32] Maghri, E.l. and Pareto-Fenchel, M. ε-subdifferential sum rule and ε-efficiency, Optim. Lett. 6 (2012), 763–781.
[33] N´emeth, A.B. A nonconvex vector minimization problem, Nonlinear Anal. 10 (1986), 669–678.
[34] Rong, W. Proper ε-efficiency in vector optimization problems with cone-subconvexlikeness, Acta Sci. Natur. Univ. NeiMongol, 28 (1997), 609–613.
[35] Shahbeyk, S. and Soleimani-damaneh, M. Proper minimal points of non-convex sets in Banach spaces in terms of the limiting normal cone, Op-timization, 66 (2017), 473–489.
[36] Shahbeyk, S. and Soleimani-damaneh, M. Limiting proper minimal points of nonconvex sets in finite-dimensional spaces, Carpathian J. Math. 35(3) (2019), 370–384.
[37] Shahbeyk, S., Soleimani-damaneh, M. and Kasimbeyli, R. Hartley prop-erly and super nondominated solutions in vector optimization with a variable ordering structure, J. Global Optim. 71 (2018), 383–405.
[38] Soleimani, B. Characterization of Approximate Solutions of Vector Op-timization Problems with a Variable Order Structure, J. Optim. Theory Appl. 162 (2014), 605–632.
[39] Soleimani, B. and Tammer, C. Concepts for approximate solutions of vector optimization problems with variable order structure, Vietnam J. Math. 42 (2014), 43–566.
[40] Tammer, C. Stability results for approximately efficient solutions, OR Spectrum, 16 (1994), 47–52.
[41] Tanaka, T. A new approach to approximation of solutions in vector optimization problems, in Proceeding of APORS, 1994, M. Fushimi and K. Tone, eds., World Scientific Publishing, Singapore, (1995) 497–504.
[42] Wacker, M. and Deinzer, F. Automatic robust medical image registration using a new democratic vector optimization approach with multiple mea-sures, in Yang GZ, Hawkes D, Rueckert D, Noble A, Taylor C, editors. Vol. 5761, Lecture notes in computer science, 12th International Con-ference, MICCAI 2009, 2009 Sept 20–24. Heidelberg, Springer, London, UK, (2009) 590–597.
[43] White, D.J. Epsilon efficiency, J. Optim. Theory Appl. 49 (1986), 319–337.
[44] Yu, P.l. Cone convexity, cone extreme points, and nondominated solu-tions in decision problems with multiobjectives, J. Optim. Theory Appl. 14 (1974), 319–377.
[45] Zamani, M. and Soleimani-damaneh, M. Proper efficiency, scalarization and transformation in multi-objective optimization: unified approaches, Optimization, (2020), 1–22.