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:
[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. ChenY. J. ChoJ. 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. JayswalR. 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. KimS. 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. KukG. 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. LiangH. 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. LiangH. 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. LongN. 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. SunejaP. 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. ChenY. J. ChoJ. 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. JayswalR. 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. KimS. 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. KukG. 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. LiangH. 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. LiangH. 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. LongN. 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. SunejaP. 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

[1]

Yubo Yuan. Canonical duality solution for alternating support vector machine. Journal of Industrial & Management Optimization, 2012, 8 (3) : 611-621. doi: 10.3934/jimo.2012.8.611

[2]

Tadeusz Antczak, Najeeb Abdulaleem. Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019089

[3]

Yubo Yuan, Weiguo Fan, Dongmei Pu. Spline function smooth support vector machine for classification. Journal of Industrial & Management Optimization, 2007, 3 (3) : 529-542. doi: 10.3934/jimo.2007.3.529

[4]

Xinmin Yang. On symmetric and self duality in vector optimization problem. Journal of Industrial & Management Optimization, 2011, 7 (3) : 523-529. doi: 10.3934/jimo.2011.7.523

[5]

Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483

[6]

Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361

[7]

Xian-Jun Long, Nan-Jing Huang, Zhi-Bin Liu. Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs. Journal of Industrial & Management Optimization, 2008, 4 (2) : 287-298. doi: 10.3934/jimo.2008.4.287

[8]

Xiao-Bing Li, Qi-Lin Wang, Zhi Lin. Optimality conditions and duality for minimax fractional programming problems with data uncertainty. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1133-1151. doi: 10.3934/jimo.2018089

[9]

Radu Ioan Boţ, Sorin-Mihai Grad. On linear vector optimization duality in infinite-dimensional spaces. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 407-415. doi: 10.3934/naco.2011.1.407

[10]

Jiawei Chen, Shengjie Li, Jen-Chih Yao. Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-18. doi: 10.3934/jimo.2018174

[11]

Fengqiu Liu, Xiaoping Xue. Subgradient-based neural network for nonconvex optimization problems in support vector machines with indefinite kernels. Journal of Industrial & Management Optimization, 2016, 12 (1) : 285-301. doi: 10.3934/jimo.2016.12.285

[12]

Ning Lu, Ying Liu. Application of support vector machine model in wind power prediction based on particle swarm optimization. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1267-1276. doi: 10.3934/dcdss.2015.8.1267

[13]

Tran Ninh Hoa, Ta Duy Phuong, Nguyen Dong Yen. Linear fractional vector optimization problems with many components in the solution sets. Journal of Industrial & Management Optimization, 2005, 1 (4) : 477-486. doi: 10.3934/jimo.2005.1.477

[14]

Anurag Jayswala, Tadeusz Antczakb, Shalini Jha. Second order modified objective function method for twice differentiable vector optimization problems over cone constraints. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 133-145. doi: 10.3934/naco.2019010

[15]

K. Schittkowski. Optimal parameter selection in support vector machines. Journal of Industrial & Management Optimization, 2005, 1 (4) : 465-476. doi: 10.3934/jimo.2005.1.465

[16]

Hong-Gunn Chew, Cheng-Chew Lim. On regularisation parameter transformation of support vector machines. Journal of Industrial & Management Optimization, 2009, 5 (2) : 403-415. doi: 10.3934/jimo.2009.5.403

[17]

Yuhua Sun, Laisheng Wang. Optimality conditions and duality in nondifferentiable interval-valued programming. Journal of Industrial & Management Optimization, 2013, 9 (1) : 131-142. doi: 10.3934/jimo.2013.9.131

[18]

Adela Capătă. Optimality conditions for vector equilibrium problems and their applications. Journal of Industrial & Management Optimization, 2013, 9 (3) : 659-669. doi: 10.3934/jimo.2013.9.659

[19]

Qiu-Sheng Qiu. Optimality conditions for vector equilibrium problems with constraints. Journal of Industrial & Management Optimization, 2009, 5 (4) : 783-790. doi: 10.3934/jimo.2009.5.783

[20]

Anulekha Dhara, Aparna Mehra. Conjugate duality for generalized convex optimization problems. Journal of Industrial & Management Optimization, 2007, 3 (3) : 415-427. doi: 10.3934/jimo.2007.3.415

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (26)
  • HTML views (285)
  • Cited by (0)

Other articles
by authors

[Back to Top]