Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T19:38:36.720Z Has data issue: false hasContentIssue false
  • English
  • Français

On generalised convex mathematical programming

Published online by Cambridge University Press:  17 February 2009

V. Jeyakumar  and
B. Mond
V. Jeyakumar
Affiliation:
School of Mathematics, University of New South Wales, Kensington, NSW, Australia2033.
B. Mond
Affiliation:
School of Mathematics and Information Sciences, La Trobe University, Vic.Australia3083.
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 sufficient optimality conditions and duality results have recently been given for the following generalised convex programming problem:

where the funtion f and g satisfy

for some η: X0 × X0 → ℝn

It is shown here that a relaxation defining the above generalised convexity leads to a new class of multi-objective problems which preserves the sufficient optimality and duality results in the scalar case, and avoids the major difficulty of verifying that the inequality holds for the same function η(. , .). Further, this relaxation allows one to treat certain nonlinear multi-objective fractional programming problems and some other classes of nonlinear (composite) problems as special cases.

Type
Research Article
Information
The ANZIAM Journal , Volume 34 , Issue 1 , July 1992 , pp. 43 - 53
Copyright
Copyright © Australian Mathematical Society 1992

References

[1] Ben-Israel, A. and Mond, B., “What is invexity?”,J. Austral. Math. Soc. Ser. B 28 (1986) 1–9.CrossRefGoogle Scholar
[2] Chandra, S., “Strong pseudo-convex programming”, Indian J. Pure Appl. Math. 3 (1972) 278–282.Google Scholar
[3] Chew, K. L. and Choo, E. U., “Pseudolinearity and efficiency”, Mathematical Programming 28 (1984) 226–239.CrossRefGoogle Scholar
[4] Craven, B. D., “Invex functions and constrained local minima”, Bull. Austral. Math. Soc. 24 (1981) 357–366.Google Scholar
[5] Craven, B. D., “Duality for generalised convex fractional programs”, in Generalised concavity in Optimisation and Economics (eds. Schaible, S. and Ziemba, W. T.) (Academic Press, New York, 1981) 473–490.Google Scholar
[6] Craven, B. D., Fractional programming, (Heldermann Verlag, Berlin, 1988).Google Scholar
[7] Craven, B. D., Mathematical programming and control theory, (Chapman and Hall, London, 1978).CrossRefGoogle Scholar
[8] Craven, B. D. and Glover, B. M., “Invex functions and duality”, J. Austral. Math. Soc. Ser. A 39 (1985) 1–20.Google Scholar
[9] Hanson, M. A., “On sufficiency of the Kuhn-Tucker conditions”, J. Math. Anal. Appl. 80 (1981) 544–550.CrossRefGoogle Scholar
[10] Martin, D. H., “The essence of invexity”, J. Optim. Theore. Appl. 47 (1985) 65–76.CrossRefGoogle Scholar
[11] Mond, B., “Generalised convexity in mathematical programming”, Bull. Austral. Math. Soc. 27 (1983) 185–202.Google Scholar
[12] Mond, B. and Weir, T., “Duality for fractional programming with generalised convexity conditions”, J. Inform. Optim. 3 (1982) 105–124.Google Scholar
[13] Weir, T., Mond, B. and Craven, B. D., “On duality for weakly minimized vector valued optimisation problems”, Optimisation 17 (1986) 711–721.CrossRefGoogle Scholar