2011, 1(3): 361-370. doi: 10.3934/naco.2011.1.361

Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity

1. 

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067

2. 

Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China

Received  April 2011 Revised  June 2011 Published  September 2011

Using a parametric approach, we establish necessary and sufficient optimality conditions and derive some duality theorems for a class of nonsmooth minmax fractional programming problems containing generalized univex functions. The results obtained in this paper extend and improve some corresponding results in the literature.
Citation: 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
References:
[1]

C. R. Bector, S. Chandra and M. K. Bector, Generalized fractional programming duality: a parametric approach,, J. Optim. Theory Appl., 60 (1989), 243. doi: 10.1007/BF00940006. Google Scholar

[2]

C. R. Bector, S. K. Duneja and S. Gupta, Univex functions and univex nonlinear programming,, in: Proceedings of the Asministrative Sciences Association of Canada, (1992), 115. Google Scholar

[3]

S. Chandra, B. D. Craven and B. Mond, Generalized fractional programming duality: a ratio game approach,, J. Austral. Math. Soc., 28 (1986), 170. doi: 10.1017/S0334270000005282. Google Scholar

[4]

J. P. Crouzeix, J. A. Ferland and S. Schaible, Duality in generalized fractional programming,, Math. Programming, 27 (1983), 342. doi: 10.1007/BF02591908. Google Scholar

[5]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", Wiley-Interscience, (1983). Google Scholar

[6]

M. A. Hanson, On sufficiency of the Kuhn-Tucker condition,, J. Math. Anal. Appl., 80 (1981), 545. doi: 10.1016/0022-247X(81)90123-2. Google Scholar

[7]

I. Husain, M. A. Hanson and Z. Jabeen, On nondifferentiable fractional minimax programming,, European J. Oper. Res., 160 (2005), 202. doi: 10.1016/S0377-2217(03)00437-5. Google Scholar

[8]

V. Jeyakumar, Equivalence of saddle-points and optima, and duality for a class of nonsmooth non-convex problems,, J. Math. Anal. Appl., 130 (1988), 334. doi: 10.1016/0022-247X(88)90309-5. Google Scholar

[9]

V. Jeyakumar and B. Mond, On generalized convex mathematical programming,, J. Austral. Math. Soc., 34 (1992), 43. doi: 10.1017/S0334270000007372. Google Scholar

[10]

G. M. Lee, Nonsmooth invexity in multiobjective programming,, J. Inform. Optim. Soc., 15 (1994), 127. Google Scholar

[11]

Z. A. Liang and Z. W. Shi, Optimality conditions and duality for a minimax fractional programming with generalized convexity,, J. Math. Anal. Appl., 277 (2003), 474. doi: 10.1016/S0022-247X(02)00553-X. Google Scholar

[12]

J. C. Liu, Optimality and duality for generalized fractional programming involving nonsmooth $(F,\rho)$-convex functions,, Comput. Math. Appl., 32 (1996), 91. doi: 10.1016/0898-1221(96)00106-X. Google Scholar

[13]

J. C. Liu, Optimality and duality for generalized fractional programming involving nonsmooth pseudoinvex functions,, J. Math. Anal. Appl., 202 (1996), 667. doi: 10.1006/jmaa.1996.0341. Google Scholar

[14]

H. Z. Luo and H. X. Wu, On necessary conditions for a class of nondifferentiable minimax fractional programming,, J. Comput. Appl. Math., 215 (2008), 103. doi: 10.1016/j.cam.2007.03.032. Google Scholar

[15]

S. K. Mishra, R. P. Pant and J. S. Rautela, Generalized $\alpha$-invexity and nondifferentiable minimax fractional programming,, J. Comput. Appl. Math., 206 (2007), 122. doi: 10.1016/j.cam.2006.06.009. Google Scholar

[16]

S. K. Mishra, S. Y. Wang, K. K. Lai and J. M. Shi, Nondifferentiable minimax fractional programming under generalized univexity,, J. Comput. Appl. Math., 158 (2007), 379. doi: 10.1016/S0377-0427(03)00455-2. Google Scholar

[17]

B. Mond and T. Weir, Generalized concavity and duality,, in, (1981), 263. Google Scholar

[18]

T. W. Reiland, Nonsmooth invexity,, Bull. Austral. Math. Soc., 42 (1990), 437. doi: 10.1017/S0004972700028604. Google Scholar

[19]

W. E. Schmitendorf, Necessary conditions and sufficient conditions for static minimax problems,, J. Math. Anal. Appl., 57 (1977), 683. doi: 10.1016/0022-247X(77)90255-4. Google Scholar

[20]

G. J. Zalmai, Optimality conditions and duality models for generalized fractional programming problems containing locally subdifferentiable and $\rho$-convex functions,, Optimization, 32 (1995), 95. doi: 10.1080/02331939508844040. Google Scholar

show all references

References:
[1]

C. R. Bector, S. Chandra and M. K. Bector, Generalized fractional programming duality: a parametric approach,, J. Optim. Theory Appl., 60 (1989), 243. doi: 10.1007/BF00940006. Google Scholar

[2]

C. R. Bector, S. K. Duneja and S. Gupta, Univex functions and univex nonlinear programming,, in: Proceedings of the Asministrative Sciences Association of Canada, (1992), 115. Google Scholar

[3]

S. Chandra, B. D. Craven and B. Mond, Generalized fractional programming duality: a ratio game approach,, J. Austral. Math. Soc., 28 (1986), 170. doi: 10.1017/S0334270000005282. Google Scholar

[4]

J. P. Crouzeix, J. A. Ferland and S. Schaible, Duality in generalized fractional programming,, Math. Programming, 27 (1983), 342. doi: 10.1007/BF02591908. Google Scholar

[5]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", Wiley-Interscience, (1983). Google Scholar

[6]

M. A. Hanson, On sufficiency of the Kuhn-Tucker condition,, J. Math. Anal. Appl., 80 (1981), 545. doi: 10.1016/0022-247X(81)90123-2. Google Scholar

[7]

I. Husain, M. A. Hanson and Z. Jabeen, On nondifferentiable fractional minimax programming,, European J. Oper. Res., 160 (2005), 202. doi: 10.1016/S0377-2217(03)00437-5. Google Scholar

[8]

V. Jeyakumar, Equivalence of saddle-points and optima, and duality for a class of nonsmooth non-convex problems,, J. Math. Anal. Appl., 130 (1988), 334. doi: 10.1016/0022-247X(88)90309-5. Google Scholar

[9]

V. Jeyakumar and B. Mond, On generalized convex mathematical programming,, J. Austral. Math. Soc., 34 (1992), 43. doi: 10.1017/S0334270000007372. Google Scholar

[10]

G. M. Lee, Nonsmooth invexity in multiobjective programming,, J. Inform. Optim. Soc., 15 (1994), 127. Google Scholar

[11]

Z. A. Liang and Z. W. Shi, Optimality conditions and duality for a minimax fractional programming with generalized convexity,, J. Math. Anal. Appl., 277 (2003), 474. doi: 10.1016/S0022-247X(02)00553-X. Google Scholar

[12]

J. C. Liu, Optimality and duality for generalized fractional programming involving nonsmooth $(F,\rho)$-convex functions,, Comput. Math. Appl., 32 (1996), 91. doi: 10.1016/0898-1221(96)00106-X. Google Scholar

[13]

J. C. Liu, Optimality and duality for generalized fractional programming involving nonsmooth pseudoinvex functions,, J. Math. Anal. Appl., 202 (1996), 667. doi: 10.1006/jmaa.1996.0341. Google Scholar

[14]

H. Z. Luo and H. X. Wu, On necessary conditions for a class of nondifferentiable minimax fractional programming,, J. Comput. Appl. Math., 215 (2008), 103. doi: 10.1016/j.cam.2007.03.032. Google Scholar

[15]

S. K. Mishra, R. P. Pant and J. S. Rautela, Generalized $\alpha$-invexity and nondifferentiable minimax fractional programming,, J. Comput. Appl. Math., 206 (2007), 122. doi: 10.1016/j.cam.2006.06.009. Google Scholar

[16]

S. K. Mishra, S. Y. Wang, K. K. Lai and J. M. Shi, Nondifferentiable minimax fractional programming under generalized univexity,, J. Comput. Appl. Math., 158 (2007), 379. doi: 10.1016/S0377-0427(03)00455-2. Google Scholar

[17]

B. Mond and T. Weir, Generalized concavity and duality,, in, (1981), 263. Google Scholar

[18]

T. W. Reiland, Nonsmooth invexity,, Bull. Austral. Math. Soc., 42 (1990), 437. doi: 10.1017/S0004972700028604. Google Scholar

[19]

W. E. Schmitendorf, Necessary conditions and sufficient conditions for static minimax problems,, J. Math. Anal. Appl., 57 (1977), 683. doi: 10.1016/0022-247X(77)90255-4. Google Scholar

[20]

G. J. Zalmai, Optimality conditions and duality models for generalized fractional programming problems containing locally subdifferentiable and $\rho$-convex functions,, Optimization, 32 (1995), 95. doi: 10.1080/02331939508844040. Google Scholar

[1]

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

[2]

Xiuhong Chen, Zhihua Li. On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity. Journal of Industrial & Management Optimization, 2018, 14 (3) : 895-912. doi: 10.3934/jimo.2017081

[3]

Ram U. Verma. General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 333-339. doi: 10.3934/naco.2011.1.333

[4]

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

[5]

Mansoureh Alavi Hejazi, Soghra Nobakhtian. Optimality conditions for multiobjective fractional programming, via convexificators. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-9. doi: 10.3934/jimo.2018170

[6]

Anurag Jayswal, Ashish Kumar Prasad, Izhar Ahmad. On minimax fractional programming problems involving generalized $(H_p,r)$-invex functions. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1001-1018. doi: 10.3934/jimo.2014.10.1001

[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]

Qinghong Zhang, Gang Chen, Ting Zhang. Duality formulations in semidefinite programming. Journal of Industrial & Management Optimization, 2010, 6 (4) : 881-893. doi: 10.3934/jimo.2010.6.881

[9]

Jen-Yen Lin, Hui-Ju Chen, Ruey-Lin Sheu. Augmented Lagrange primal-dual approach for generalized fractional programming problems. Journal of Industrial & Management Optimization, 2013, 9 (4) : 723-741. doi: 10.3934/jimo.2013.9.723

[10]

Mehar Chand, Jyotindra C. Prajapati, Ebenezer Bonyah, Jatinder Kumar Bansal. Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 539-560. doi: 10.3934/dcdss.2020030

[11]

Mohamed Aly Tawhid. Nonsmooth generalized complementarity as unconstrained optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 411-423. doi: 10.3934/jimo.2010.6.411

[12]

Yanqun Liu. Duality in linear programming: From trichotomy to quadrichotomy. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1003-1011. doi: 10.3934/jimo.2011.7.1003

[13]

Xinmin Yang. On second order symmetric duality in nondifferentiable multiobjective programming. Journal of Industrial & Management Optimization, 2009, 5 (4) : 697-703. doi: 10.3934/jimo.2009.5.697

[14]

Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018140

[15]

Gang Li, Lipu Zhang, Zhe Liu. The stable duality of DC programs for composite convex functions. Journal of Industrial & Management Optimization, 2017, 13 (1) : 63-79. doi: 10.3934/jimo.2016004

[16]

Regina Sandra Burachik, Alex Rubinov. On the absence of duality gap for Lagrange-type functions. Journal of Industrial & Management Optimization, 2005, 1 (1) : 33-38. doi: 10.3934/jimo.2005.1.33

[17]

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

[18]

Kamil Otal, Ferruh Özbudak, Wolfgang Willems. Self-duality of generalized twisted Gabidulin codes. Advances in Mathematics of Communications, 2018, 12 (4) : 707-721. doi: 10.3934/amc.2018042

[19]

Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295

[20]

Majid E. Abbasov. Generalized exhausters: Existence, construction, optimality conditions. Journal of Industrial & Management Optimization, 2015, 11 (1) : 217-230. doi: 10.3934/jimo.2015.11.217

 Impact Factor: 

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]