Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T13:08:34.783Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalised convexity and duality in multiple objective programming

Published online by Cambridge University Press:  17 April 2009

T. Weir  and
B. Mond
T. Weir
Affiliation:
7/35 Gaza Rd., West Ryde NSW 2114Australia
B. Mond
Affiliation:
Department of MathematicsLa Trobe UniversityBundoora, Victoria, 3083Australia
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.

By considering the concept of weak minima, different scalar duality results are extended to multiple objective programming problems. A number of weak, strong and converse duality theorems are given under a variety of generalised convexity conditions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 39 , Issue 2 , April 1989 , pp. 287 - 299
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Bector, C.R.Bector, M.K. and Klassen, J.E., ‘Duality for a nonlinear programming problem’, Utilitas Math 11 (1977), 87–99.Google Scholar
[2]Bector, C.R. and Bector, M.K., ‘On various duality theorems in nonlinear programming’, J. Optim. Theory Appl 53 (1987), 509–515.CrossRefGoogle Scholar
[3]Bitran, G., ‘Duality in nonlinear multiple criteria optimization problems’, J. Optim. Theory Appl. 35 (1981), 367–406.CrossRefGoogle Scholar
[4]Corley, B.D., ‘Duality theory for maximizations with respect to cones’, J. Math. Anal. Appl 84 (1981), 560–568.CrossRefGoogle Scholar
[5]Craven, B.D. and Mond, B., ‘On converse duality in nonlinear programming’, Oper. Res 19 (1971), 1075–1078.CrossRefGoogle Scholar
[6]Craven, B.D., ‘Langrangian conditions and quasiduality’, Bull. Austral. Math. Soc 16 (1977), 325–339.CrossRefGoogle Scholar
[7]Craven, B.D., ‘Strong vector minimization and duality’, Z. Angew. Math. Mech 60 (1980), 1–5.CrossRefGoogle Scholar
[8]Geoffrion, A.M., ‘Proper efficiency and the theory of vector maximization’, J. Math. Anal. Appl 22 (1968), 618–630.CrossRefGoogle Scholar
[9]Ivanov, E. H. and Nehse, R., ‘Some results on dual vector optimization problems’, Optimization 16 (1985), 505–517.CrossRefGoogle Scholar
[10]Mahajan, D. G. and Vartak, M.N., ‘Generalization of some duality theorems in nonlinear programming’, Math. Programming 12 (1977), 293–317.CrossRefGoogle Scholar
[11]Mangasarian, O. L., Nonlinear programming (McGraw-Hill, New York, 1969).Google Scholar
[12]Mond, B. and Weir, T., ‘Generalized concavity and duality’, in Generalized concavity in optimization and economics, Edited by Schaible, S. and Ziemba, W.T., pp. 263–279 (Academic Press, New York, 1981).Google Scholar
[13]Tanino, T. and Sawaragi, Y., ‘Duality theory in multiobjective programming’, J. Optim. Theory Appl 27 (1979), 509–529.CrossRefGoogle Scholar
[14]Weir, T., ‘Generalized Convexity and Duality in Mathematical Programming’, Ph.D. Thesis. (La Trobe University, Bundoora, Melbourne, Australia, 1982).Google Scholar
[15]Weir, T., ‘A note on strict converse duality in nonlinear programming’, J. Inf. Opt. Sci 7 (1986), 65–71.Google Scholar
[16]Weir, T., ‘Proper efficiency and duality for vector valued optimization problems’, J. Austral. Math. Soc., Series A 43 (1987), 21–34.CrossRefGoogle Scholar
[17]White, D. G., ‘Vector maximization and Lagrange multipliers’, Math. Programming 31 (1985), 192–205.CrossRefGoogle Scholar
[18]Wolfe, P., ‘A duality theorem for nonlinear programming’, Quart. Appl. Math 19 (1961), 239–244.CrossRefGoogle Scholar