# American Institute of Mathematical Sciences

April  2017, 13(2): 549-572. doi: 10.3934/jimo.2016031

## On fractional vector optimization over cones with support functions

 Department of Mathematics, University of Delhi, Delhi-110 007, India

Received  December 2013 Revised  October 2015 Published  May 2016

In this paper we give necessary and sufficient optimality conditions for a fractional vector optimization problem over cones involving support functions in objective as well as constraints, using cone-convex functions. We also associate Mond-Weir type and Schaible type duals with the primal problem and establish weak and strong duality results under cone-convexity, pseudoconvexity and quasiconvexity assumptions. A number of previously studied problems appear as special cases.

Citation: Pooja Louhan, S. K. Suneja. On fractional vector optimization over cones with support functions. Journal of Industrial & Management Optimization, 2017, 13 (2) : 549-572. doi: 10.3934/jimo.2016031
##### References:
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T. Luc, Theory of Vector Optimization, Springer, 1989. Google Scholar [20] B. Mond and M. Schechter, A duality theorem for a homogeneous fractional programming problem, Journal of Optimization Theory and Applications, 25 (1978), 349-359. doi: 10.1007/BF00932898. Google Scholar [21] S. Schaible, Fractional programming Ⅰ: Duality, Management Science, 22 (1975/76), 858-867. doi: 10.1287/mnsc.22.8.858. Google Scholar [22] S. Schaible, Fractional programming Ⅱ: On Dinkelbach's algorithm., Management Science, 22 (1975/76), 868-873. doi: 10.1287/mnsc.22.8.868. Google Scholar [23] S. Schaible and T. Ibaraki, Fractional programming, European Journal of Operational Research, 12 (1983), 325-338. doi: 10.1016/0377-2217(83)90153-4. Google Scholar [24] S. K. Suneja and S. Gupta, Duality in multiple objective fractional programming problems involving non-convex functions, OPSEARCH, 27 (1990), 239-253. Google Scholar [25] S. K. Suneja, P. Louhan and M. B. Grover, Higher-order cone-pseudoconvex, quasiconvex and other related functions in vector optimization, Optimization Letters, 7 (2013), 647-664. doi: 10.1007/s11590-012-0447-y. Google Scholar [26] G. J. Zalmai, Generalized $(η, ρ)$-invex functions and global semiparametric sufficient efficiency conditions for multiobjective fractional programming problems containing arbitrary norms, Journal of Global Optimization, 36 (2006), 51-85. doi: 10.1007/s10898-006-6586-x. Google Scholar

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##### References:
 [1] T. Antczak, A modified objective function method for solving nonlinear multiobjective fractional programming problems, Journal of Mathematical Analysis and Applications, 322 (2006), 971-989. doi: 10.1016/j.jmaa.2005.08.098. Google Scholar [2] R. Cambini, Some new classes of generalized concave vector-valued functions, Optimization, 36 (1996), 11-24. doi: 10.1080/02331939608844161. Google Scholar [3] A. Charnes and W. W. Cooper, Programming with linear fractional functionals, Naval Research Logistic Quarterly, 9 (1962), 181-186. doi: 10.1002/nav.3800090303. Google Scholar [4] J. W. Chen, Y. J. Cho, J. K. Kim and J. Li, Multiobjective optimization problems with modified objective functions and cone constraints and applications, Journal of Global Optimization, 49 (2011), 137-147. doi: 10.1007/s10898-010-9539-3. Google Scholar [5] F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, 1983. Google Scholar [6] B. D. Craven, Nonsmooth multiobjective programming, Numerical Functional Analysis and Optimization, 10 (1989), 49-64. doi: 10.1080/01630568908816290. Google Scholar [7] W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498. doi: 10.1287/mnsc.13.7.492. Google Scholar [8] I. Husain and Z. Jabeen, On fractional programming containing support functions, Journal of Applied Mathematics and Computing, 18 (2005), 361-376. doi: 10.1007/BF02936579. Google Scholar [9] A. Jayswal, R. Kumar and D. Kumar, Multiobjective fractional programming problems involving $(p,r)$-$ρ$-$(η,θ)$-invex function, Journal of Applied Mathematics and Computing, 39 (2012), 35-51. doi: 10.1007/s12190-011-0508-x. Google Scholar [10] D. S. Kim, Multiobjective fractional programming with a modified objective function, Communications of the Korean Mathematical Society, 20 (2005), 837-847. doi: 10.4134/CKMS.2005.20.4.837. Google Scholar [11] D. S. Kim, Nonsmooth multiobjective fractional programming with generalized invexity, Taiwanese Journal of Mathematics, 10 (2006), 467-478. Google Scholar [12] D. S. Kim, S. J. Kim and M. H. Kim, Optimality and duality for a class of nondifferentiable multiobjective fractional programming problems, Journal of Optimization Theory and Applications, 129 (2006), 131-146. doi: 10.1007/s10957-006-9048-1. Google Scholar [13] M. H. Kim and G. S. Kim, On optimality and duality for generalized nondifferentiable fractional optimization problems, Communications of the Korean Mathematical Society, 25 (2010), 139-147. doi: 10.4134/CKMS.2010.25.1.139. Google Scholar [14] H. Kuk, G. M. Lee and T. Tanino, Optimality and duality for nonsmooth multiobjective fractional programming with generalized invexity, Journal of Mathematical Analysis and Applications, 262 (2001), 365-375. doi: 10.1006/jmaa.2001.7586. Google Scholar [15] Z. A. Liang, H. X. Huang and P. M. Pardalos, Optimality conditions and duality for a class of nonlinear fractional programming problems, Journal of Optimization Theory and Applications, 110 (2001), 611-619. doi: 10.1023/A:1017540412396. Google Scholar [16] Z. A. Liang, H. X. Huang and P. M. Pardalos, Efficiency conditions and duality for a class of multiobjective fractional programming problems, Journal of Global Optimization, 27 (2003), 447-471. doi: 10.1023/A:1026041403408. Google Scholar [17] X. J. Long, Optimality conditions and duality for nondifferentiable multiobjective fractional programming problems with $(C,α,ρ,d)$-convexity, Journal of Optimization Theory and Applications, 148 (2011), 197-208. doi: 10.1007/s10957-010-9740-z. Google Scholar [18] X. J. Long, N. J. Huang and Z. B. Liu, Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs, Journal of Industrial and Management Optimization, 4 (2008), 287-298. doi: 10.3934/jimo.2008.4.287. Google Scholar [19] D. T. Luc, Theory of Vector Optimization, Springer, 1989. Google Scholar [20] B. Mond and M. Schechter, A duality theorem for a homogeneous fractional programming problem, Journal of Optimization Theory and Applications, 25 (1978), 349-359. doi: 10.1007/BF00932898. Google Scholar [21] S. Schaible, Fractional programming Ⅰ: Duality, Management Science, 22 (1975/76), 858-867. doi: 10.1287/mnsc.22.8.858. Google Scholar [22] S. Schaible, Fractional programming Ⅱ: On Dinkelbach's algorithm., Management Science, 22 (1975/76), 868-873. doi: 10.1287/mnsc.22.8.868. Google Scholar [23] S. Schaible and T. Ibaraki, Fractional programming, European Journal of Operational Research, 12 (1983), 325-338. doi: 10.1016/0377-2217(83)90153-4. Google Scholar [24] S. K. Suneja and S. Gupta, Duality in multiple objective fractional programming problems involving non-convex functions, OPSEARCH, 27 (1990), 239-253. Google Scholar [25] S. K. Suneja, P. Louhan and M. B. Grover, Higher-order cone-pseudoconvex, quasiconvex and other related functions in vector optimization, Optimization Letters, 7 (2013), 647-664. doi: 10.1007/s11590-012-0447-y. Google Scholar [26] G. J. Zalmai, Generalized $(η, ρ)$-invex functions and global semiparametric sufficient efficiency conditions for multiobjective fractional programming problems containing arbitrary norms, Journal of Global Optimization, 36 (2006), 51-85. doi: 10.1007/s10898-006-6586-x. Google Scholar
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