J Austral Math Soc Ser B 34 pp43--53, 1992.

On generalised convex mathematical programming

V. Jeyakumar and B. Mond

(Received 22 February 1991; revised 21 March 1991)

Abstract

The sufficient optimality conditions and duality results have recently been given for the following generalised convex programming problem:

Minimise f (x) , subject to g(x) £ 0 , x Î X0 Ì Rn ,
where the functions f and g satisfy

ì f (x) - f (a) - f ' (a)h(x, a) ³ 0  
x, a Î X0 Þ í                                                                     
î g(x) - g(a) - g ' (a)h(x, a) ³ 0 ,

for some h: X0 ´ X0 Î 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 h(., .). 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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Authors

V. Jeyakumar
School of Mathematics, University of New SOuth Wales, Kensington, NSW, Australia 2033.
B. Mond
School of Mathematics and Information Sciences, La Trobe University, Vic. Australia 3083.