On fractional vector optimization over cones with support functions

Pooja Louhan; S. K. Suneja

doi:10.3934/jimo.2016031

Article Contents

2017, Volume 13, Issue 2: 549-572. Doi: 10.3934/jimo.2016031

This issue Previous Article The linear convergence of a derivative-free descent method for nonlinear complementarity problems Next Article A parametric simplex algorithm for biobjective piecewise linear programming problems

On fractional vector optimization over cones with support functions

Pooja Louhan and
S. K. Suneja

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

Received: November 30, 2013

Revised: September 30, 2015

Published: August 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 90C29; Secondary: 90C46, 90C25, 90C26.

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[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.
[2]	R. Cambini, Some new classes of generalized concave vector-valued functions, Optimization, 36 (1996), 11-24. doi: 10.1080/02331939608844161.
[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.
[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.
[5]	F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, 1983.
[6]	B. D. Craven, Nonsmooth multiobjective programming, Numerical Functional Analysis and Optimization, 10 (1989), 49-64. doi: 10.1080/01630568908816290.
[7]	W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498. doi: 10.1287/mnsc.13.7.492.
[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.
[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.
[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.
[11]	D. S. Kim, Nonsmooth multiobjective fractional programming with generalized invexity, Taiwanese Journal of Mathematics, 10 (2006), 467-478.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	D. T. Luc, Theory of Vector Optimization, Springer, 1989.
[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.
[21]	S. Schaible, Fractional programming Ⅰ: Duality, Management Science, 22 (1975/76), 858-867. doi: 10.1287/mnsc.22.8.858.
[22]	S. Schaible, Fractional programming Ⅱ: On Dinkelbach's algorithm., Management Science, 22 (1975/76), 868-873. doi: 10.1287/mnsc.22.8.868.
[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.
[24]	S. K. Suneja and S. Gupta, Duality in multiple objective fractional programming problems involving non-convex functions, OPSEARCH, 27 (1990), 239-253.
[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.
[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.