Hostname: page-component-7b7567c8c-jswmv Total loading time: 0 Render date: 2026-11-15T21:58:04.721Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Information

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