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On generalised convex mathematical programming

  • V. Jeyakumar (a1) and B. Mond (a2)

Abstract

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.

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Copyright

References

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[1] Ben-Israel, A. and Mond, B., “What is invexity?”,J. Austral. Math. Soc. Ser. B 28 (1986) 19.
[2] Chandra, S., “Strong pseudo-convex programming”, Indian J. Pure Appl. Math. 3 (1972) 278282.
[3] Chew, K. L. and Choo, E. U., “Pseudolinearity and efficiency”, Mathematical Programming 28 (1984) 226239.
[4] Craven, B. D., “Invex functions and constrained local minima”, Bull. Austral. Math. Soc. 24 (1981) 357366.
[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) 473490.
[6] Craven, B. D., Fractional programming, (Heldermann Verlag, Berlin, 1988).
[7] Craven, B. D., Mathematical programming and control theory, (Chapman and Hall, London, 1978).
[8] Craven, B. D. and Glover, B. M., “Invex functions and duality”, J. Austral. Math. Soc. Ser. A 39 (1985) 120.
[9] Hanson, M. A., “On sufficiency of the Kuhn-Tucker conditions”, J. Math. Anal. Appl. 80 (1981) 544550.
[10] Martin, D. H., “The essence of invexity”, J. Optim. Theore. Appl. 47 (1985) 6576.
[11] Mond, B., “Generalised convexity in mathematical programming”, Bull. Austral. Math. Soc. 27 (1983) 185202.
[12] Mond, B. and Weir, T., “Duality for fractional programming with generalised convexity conditions”, J. Inform. Optim. 3 (1982) 105124.
[13] Weir, T., Mond, B. and Craven, B. D., “On duality for weakly minimized vector valued optimisation problems”, Optimisation 17 (1986) 711721.
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The ANZIAM Journal
  • ISSN: 1446-1811
  • EISSN: 1446-8735
  • URL: /core/journals/anziam-journal
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