Set-valued minimax fractional programming problems under ρ-cone arcwise connectedness

Das, Koushik

doi:10.2478/candc-2022-0004

Abstract

In this paper, we consider a set-valued minimax fractional programming problem (MFP), where the objective as well as constraint maps are set-valued. We introduce the notion of ρ-cone arcwise connectedness of set-valued maps as a generalization of cone arcwise connected set-valued maps. We establish the sufficient Karush-Kuhn-Tucker (KKT) conditions for the existence of minimizers of the problem (MFP) under ρ-cone arcwise connectedness assumption. Further, we study the Mond-Weir (MWD), Wolfe (WD), and mixed (MD) types of duality models and prove the corresponding weak, strong, and converse duality theorems between the primal (MFP) and the corresponding dual problems under ρ-cone arcwise connectedness assumption.

References

AHMAD, I. (2003) optimality conditions and duality in fractional minimax programming Involving generalized ρ-invexity. internat. J. Manag. Syst. 19, 165–180.
Search in Google Scholar Back to article
AHMAD, I. and HUSAIN, Z. (2006) optimality conditions and duality in non-differentiable Minimax fractional programming with generalized convexity. J. Optim. Theory Appl. 129(2), 255–275.10.1007/s10957-006-9057-0
Search in Google Scholar Back to article
AUBIN, J.P. (1981) Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. In: L. Nachbin (ed.), Mathematical Analysis and Applications, Part A. Academic Press, New York, 160–229.
Search in Google Scholar Back to article
AUBIN, J.P. and FRANKOWSKA, H. (1990) Set-Valued Analysis. Birhäuser, Boston.
Search in Google Scholar Back to article
AVRIEL, M. (1976) Nonlinear Programming: Theory and Method. Prentice-Hall, Englewood Cliffs, New Jersey.
Search in Google Scholar Back to article
BECTOR, C.R. and BHATIA, B.L. (1985) Sufficient optimality conditions and duality for a minmax problem. Util. Math. 27, 229–247.
Search in Google Scholar Back to article
BORWEIN, J. (1977) Multivalued convexity and optimization: a unified approach to inequality and equality constraints. Math. Program. 13(1), 183–199.10.1007/BF01584336
Search in Google Scholar Back to article
CAMBINI, A., MARTEIN, L. and VLACH, M. (1999) Second order tangent sets and optimality conditions. Math. Japonica 49(3), 451–461.
Search in Google Scholar Back to article
CHANDRA, S. and KUMAR, V. (1995) Duality in fractional minimax programming. J. Austral. Math. Soc. (Ser. A.) 58, 376–386.
Search in Google Scholar Back to article
CORLEY, H.W. (1987) Existence and Lagrangian duality for maximizations of set-valued functions. J. Optim. Theory Appl. 54(3), 489–501.10.1007/BF00940198
Search in Google Scholar Back to article
DAS, K. and NAHAK, C. (2014) Sufficient optimality conditions and duality theorems for set-valued optimization problem under generalized cone convexity. Rend. Circ. Mat. Palermo 63(3), 329–345.10.1007/s12215-014-0163-9
Search in Google Scholar Back to article
DAS, K. and NAHAK, C. (2016a) Optimality conditions for approximate quasi efficiency in set-valued equilibrium problems. SeMA J. 73(2), 183–199.10.1007/s40324-016-0063-3
Search in Google Scholar Back to article
DAS, K. and NAHAK, C. (2016b) Set-valued fractional programming problems under generalized cone convexity. Opsearch 53(1), 157–177.10.1007/s12597-015-0222-9
Search in Google Scholar Back to article
DAS, K. and NAHAK, C. (2017a) Approximate quasi efficiency of set-valued optimization problems via weak subdifferential. SeMA J. 74(4), 523–542.10.1007/s40324-016-0099-4
Search in Google Scholar Back to article
DAS, K. and NAHAK, C. (2017b) Set-valued minimax programming problems under generalized cone convexity. Rend. Circ. Mat. Palermo 66(3), 361–374.10.1007/s12215-016-0258-6
Search in Google Scholar Back to article
DAS, K. and NAHAK, C. (2020a) Optimality conditions for set-valued minimax fractional programming problems. SeMA J. 77(2), 161–179.10.1007/s40324-019-00209-7
Search in Google Scholar Back to article
DAS, K. and NAHAK, C. (2020b) Optimality conditions for set-valued minimax programming problems via second-order contingent epiderivative. J. Sci. Res. 64(2), 313–321.10.37398/JSR.2020.640243
Search in Google Scholar Back to article
FU, J. and WANG, Y. (2003) Arcwise connected cone-convex functions and mathematical programming. J. Optim. Theory Appl. 118(2), 339–352.10.1023/A:1025451422581
Search in Google Scholar Back to article
JAHN, J. and RAUH, R. (1997) Contingent epiderivatives and set-valued optimization. Math. Method Oper. Res. 46(2), 193–211.10.1007/BF01217690
Search in Google Scholar Back to article
LAI, H.C. and LEE, J.C. (2002) On duality theorems for nondifferentiable minimax fractional programming. J. Comput. Appl. Math. 146, 115–126.
Search in Google Scholar Back to article
LAI, H.C., LIU, J.C. and TANAKA, K. (1999) Necessary and sufficient conditions for minimax fractional programming. J. Math. Anal. Appl. 230, 311–328.
Search in Google Scholar Back to article
LALITHA, C., DUTTA, J. and GOVIL, M.G. (2003) Optimality criteria in set-valued optimization. J. Aust. Math. Soc. 75(2), 221–232.10.1017/S1446788700003736
Search in Google Scholar Back to article
LIANG, Z.A. and SHI, Z.W. (2003) Optimality conditions and duality for minimax fractional programming with generalized convexity. J. Math. Anal. Appl. 277, 474–488.
Search in Google Scholar Back to article
LIU, J.C. and WU, C.S. (1998) On minimax fractional optimality conditions with (F, ρ)-convexity. J. Math. Anal. Appl. 219, 36–51.
Search in Google Scholar Back to article
MISHRA, S.K. (1995) Pseudolinear fractional minmax programming. Indian J. Pure Appl. Math. 26, 763–772.
Search in Google Scholar Back to article
MISHRA, S.K. (1998) Generalized pseudo convex minmax programming. Opsearch 35(1), 32–44.10.1007/BF03398537
Search in Google Scholar Back to article
MISHRA, S.K. (2001) Pseudoconvex complex minimax programming. Indian J. Pure Appl. Math. 32(2), 205–214.
Search in Google Scholar Back to article
MISHRA, S.K., WANG, S.Y. and LAI, K.K. (2004) Complex minimax programming under generalized convexity. J. Comput. Appl. Math. 167(1), 59–71.10.1016/j.cam.2003.09.045
Search in Google Scholar Back to article
MISHRA, S.K., WANG, S.Y., LAI, K.K. and SHI, J.M. (2003) Nondifferentiable minimax fractional programming under generalized univexity. J. Comput. Appl. Math. 158(2), 379–395.10.1016/S0377-0427(03)00455-2
Search in Google Scholar Back to article
PENG, Z. and XU, Y. (2018) Second-order optimality conditions for cone-subarcwise connected set-valued optimization problems. Acta Math. Appl. Sin. Engl. Ser. 34(1), 183–196.10.1007/s10255-018-0738-x
Search in Google Scholar Back to article
TREANTA, S. and DAS, K. (2021) On robust saddle-point criterion in optimization problems with curvilinear integral functionals. Mathematics 9(15), 1–13.
Search in Google Scholar Back to article
WEIR, T. (1992) Pseudoconvex minimax programming. Util. Math. 42, 234–240.
Search in Google Scholar Back to article
YADAV, S.R. and MUKHERJEE, R.N. (1990) Duality for fractional minimax programming problems. J. Austral. Math. Soc. (Ser. B.) 31, 484–492.
Search in Google Scholar Back to article
YIHONG, X. and MIN, L. (2016) Optimality conditions for weakly efficient elements of set-valued optimization with α-order near cone-arcwise connectedness. J. Systems Sci. Math. Sci. 36(10), 1721–1729.
Search in Google Scholar Back to article
YU, G. (2013) Optimality of global proper efficiency for cone-arcwise connected set-valued optimization using contingent epiderivative. Asia-Pac. J. Oper. Res. 30(03), 1340004.10.1142/S0217595913400046
Search in Google Scholar Back to article
YU, G. (2016) Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numer. Algebra, Control & Optim. 6(1), 35–44.10.3934/naco.2016.6.35
Search in Google Scholar Back to article
ZALMAI, G.J. (1987) Optimality criteria and duality for a class of minimax programming problems with generalized invexity conditions. Util. Math. 32, 35–57.
Search in Google Scholar Back to article

Set-valued minimax fractional programming problems under ρ-cone arcwise connectedness

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