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Minimax fractional programming involving generalised invex functions

Published online by Cambridge University Press:  17 February 2009

H. C. Lai  and
J. C. Liu
H. C. Lai
Affiliation:
Department of Applied Mathematics, Chung Yuang Christian University, Chung Li 320, Taiwan; e-mail: hclai@cycu.edu.tw.
J. C. Liu
Affiliation:
Section of Mathematics, National Overseas Chinese Student University, Linkou 24499 PO Box 1-1337, Taiwan.
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Abstract

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The convexity assumptions for a minimax fractional programming problem of variational type are relaxed to those of a generalised invexity situation. Sufficient optimality conditions are established under some specific assumptions. Employing the existence of a solution for the minimax variational fractional problem, three dual models, the Wolfe type dual, the Mond-Weir type dual and a one parameter dual type, are constructed. Several duality theorems concerning weak, strong and strict converse duality under the framework of invexity are proved.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 3 , January 2003 , pp. 339 - 354
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Bector, C. R., Chandra, S. and Husain, I., “Generalized continuous fractional programming duality: a parametric approach”, Util. Math. 42 (1992) 39–60.Google Scholar
[2]Bector, C. R., Chandra, S. and Kumar, V., “Duality for minimax programming involving V-invex functions”, Optimization 30 (1994) 93–103.CrossRefGoogle Scholar
[3]Bector, C. R. and Suneja, S. K., “Duality in nondifferentiable generalized fractional programming”, Asia-Pacific J. Oper Res. 5 (1988) 134–139.Google Scholar
[4]Chandra, S., Craven, B. D. and Husain, I., “A class of nondifferentiable continuous programming problems”, J. Math. Anal. Appl. 107 (1985) 122–131.CrossRefGoogle Scholar
[5]Chandra, S., Craven, B. D. and Husain, I., “Continuous programming containing arbitrary norms”, J. Austral. Math. Soc. Ser. A 39 (1985) 28–38.CrossRefGoogle Scholar
[6]Chandra, S., Craven, B. D. and Mond, B., “Generalized fractional programming duality: a ratio game approach”, J. Austral. Math. Soc. Ser. B 28 (1986) 170–180.CrossRefGoogle Scholar
[7]Craven, B. D., “Lagrangian conditions for a minimax”, in Workshop/Miniconference on Functional Analysis and Optimization (July 8–24, 1988), (Proc. Center Math. Anal., Australian National University, 1988) 24–33.Google Scholar
[8]Craven, B. D., “On continuous programming with generalized convexity”, Asia-Pacific J. Oper. Res. 10 (1993) 219–231.Google Scholar
[9]Crouzeix, J.-P., Ferland, J. A. and Schaible, S., “Duality in generalized fractional programming”, Math. Prog. 27 (1983) 342–354.CrossRefGoogle Scholar
[10]Hanson, M. A. and Mond, B., “Further generalizations of convexity in mathematical programming”, J. Inf. Optim. Sci. 3 (1982) 25–32.Google Scholar
[11]Lai, H. C. and Liu, J. C., “Optimality conditions for multiobjective programming with generalized (ℑ, ρ, θ)-convex set functions”, J. Math. Anal. Appl. 215 (1997) 443–460.CrossRefGoogle Scholar
[12]Lai, H. C. and Liu, J. C., “Duality for a minimax programming problem containing n-set functions”, J. Math. Anal. Appl. 229 (1999) 587–604.CrossRefGoogle Scholar
[13]Lai, H. C. and Liu, J. C., “On minimax fractional programming of generalized convex set functions”, J. Math. Anal. Appl. 244 (2000) 442–405.CrossRefGoogle Scholar
[14]Lai, H. C., Liu, J. C. and Tanaka, K., “Duality without a constraint qualification for minimax fractional programming”, J. Optim. Theory Appl. 101 (1999) 91–102.CrossRefGoogle Scholar
[15]Lai, H. C., Liu, J. C. and Tanaka, K., “Necessary and sufficient conditions for minimax fractional programming”, J. Math. Anal. Appl. 230 (1999) 311–328.CrossRefGoogle Scholar
[16]Liu, J. C., “Optimality and duality for multiobjective fractional programming involving nonsmooth (f, ρ)-convex functions”, Optimization 36 (1996) 333–346.CrossRefGoogle Scholar
[17]Liu, J. C., “Optimality and duality for multiobjective fractional programming involving nonsmooth (f, ρ)-convex functions”, Optimization 37 (1996) 369–383.CrossRefGoogle Scholar
[18]Mond, B., Chandra, S. and Husain, I., “Duality for variational problems with invexity”, J. Math. Anal. Appl. 134 (1988) 322–328.CrossRefGoogle Scholar
[19]Mond, B. and Husain, I., “Sufficient optimality criteria and duality for variational problems with generalized invexity”, J. Austral. Math. Soc. Ser. B 31 (1989) 108–121.CrossRefGoogle Scholar
[20]Stancu-Minasian, I. M., Fractional programming theory, methods and applications (Kluwer, Dordrecht, 1997).CrossRefGoogle Scholar
[21]Zalmai, G. J., “Optimality conditions and duality for a class of continuous-time generalized fractional programming problems”, J. Math. Anal. Appl. 153 (1990) 356–371.CrossRefGoogle Scholar
[22]Zalmai, G. J., “Optimality conditions and duality models for generalized fractional programming problems containing locally subdifferentiable and ρ-convex functions”, Optimization 32 (1995) 95–124.CrossRefGoogle Scholar