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On fractional vector optimization over cones with support functions
Department of Mathematics, University of Delhi, Delhi-110 007, India |
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.
References:
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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. |
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. |
| [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. |
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