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Minimax fractional programming involving generalised invex functions. (English) Zbl 1042.90046

The authors consider the following minimax problem with a fractional objective function.

v * =min x max 1ip a b f i (t,x(t),x ˙(t))dt a b g i (t,x(t),x ˙(t))dt

subject to xPS(T, n ),x(a)=α,x(b)=β

a b h j (t,x(t),x ˙(t))dt0,jm ̲{1,2,...,m},tT=[a,b],

where the functions f i ,g i ,ip ̲, and h j ,jm ̲ are continuous in t,x and x ˙ and have continuous partial derivatives with respect to x and x ˙, and where PS(T, n ) is the space of all piecewise smooth state functions x defined on the compact time set T in .

For this problem, sufficient optimality conditions are established in the case in which the usual convexity assumptions are relaxed to those of a generalized invexity situation. Three dual models, the Wolfe type dual, the Mond-Weir type dual and a one parameter dual type are formulated, and weak, strong and strict converse duality theorems are proved.

MSC:
90C32Fractional programming
90C47Minimax problems