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

The authors consider the following minimax problem with a fractional objective function. $v^{\ast }=\min_{x}\max_{1\leq i\leq p}\frac{ \int_{a}^{b}f^{i}( t,x(t),\dot{x}(t))\,dt}{ \int_{a}^{b}g^{i}( t,x(t),\dot{x}(t))\,dt}$ subject to $$x\in PS(T,\mathbb{R}^{n}),~x(a)=\alpha ,~x(b)=\beta$$ $\int_{a}^{b}h^{j}( t,x(t),\dot{x}(t))\,dt\leq 0, \quad j\in \underline{m}\equiv \{ 1,2,\ldots ,m\} ,~t\in T=[ a,b ],$ where the functions $$f^{i},~g^{i},~i\in \underline{p},$$and $$h^{j},~j\in \underline{m}$$ are continuous in $$t,x$$and $$\dot{x}$$and have continuous partial derivatives with respect to $$x$$and $$\dot{x}$$, and where $$PS(T,\mathbb{R}^{n})$$ is the space of all piecewise smooth state functions $$x$$ defined on the compact time set $$T$$ in $${\mathbb{R}}$$.
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:

 90C32 Fractional programming 90C47 Minimax problems in mathematical programming
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### References:

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