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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.

${v}^{*}=\underset{x}{min}\underset{1\le i\le p}{max}\frac{{\int }_{a}^{b}{f}^{i}\left(t,x\left(t\right),\stackrel{˙}{x}\left(t\right)\right)\phantom{\rule{0.166667em}{0ex}}dt}{{\int }_{a}^{b}{g}^{i}\left(t,x\left(t\right),\stackrel{˙}{x}\left(t\right)\right)\phantom{\rule{0.166667em}{0ex}}dt}$

subject to $x\in PS\left(T,{ℝ}^{n}\right),\phantom{\rule{3.33333pt}{0ex}}x\left(a\right)=\alpha ,\phantom{\rule{3.33333pt}{0ex}}x\left(b\right)=\beta$

${\int }_{a}^{b}{h}^{j}\left(t,x\left(t\right),\stackrel{˙}{x}\left(t\right)\right)\phantom{\rule{0.166667em}{0ex}}dt\le 0,\phantom{\rule{1.em}{0ex}}j\in \underline{m}\equiv \left\{1,2,...,m\right\},\phantom{\rule{3.33333pt}{0ex}}t\in T=\left[a,b\right],$

where the functions ${f}^{i},\phantom{\rule{3.33333pt}{0ex}}{g}^{i},\phantom{\rule{3.33333pt}{0ex}}i\in \underline{p},$ and ${h}^{j},\phantom{\rule{3.33333pt}{0ex}}j\in \underline{m}$ are continuous in $t,x$ and $\stackrel{˙}{x}$ and have continuous partial derivatives with respect to $x$ and $\stackrel{˙}{x}$, and where $PS\left(T,{ℝ}^{n}\right)$ is the space of all piecewise smooth state functions $x$ defined on the compact time set $T$ in $ℝ$.

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