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

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  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.

Keywords: Fractional vector optimization, cones, support function, optimality, duality.
Mathematics Subject Classification: Primary: 90C29; Secondary: 90C46, 90C25, 90C26.
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:
[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

show all references

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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