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Zbl 1042.90046
Lai, H. C.; Liu, J. C.
Minimax fractional programming involving generalised invex functions.
(English)
[J] ANZIAM J. 44, No. 3, 339-354 (2003). ISSN 1446-1811

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 $x\in PS(T,\bbfR^{n}),~x(a)=\alpha ,~x(b)=\beta$ $$\int_{a}^{b}h^{j}( t,x(t),\dot{x}(t))\,dt\leq 0, \quad j\in \underline{m}\equiv \{ 1,2,\ldots ,m\} ,~t\in T=[ a,b ],$$ 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 $PS(T,\bbfR^{n})$ is the space of all piecewise smooth state functions $x$ defined on the compact time set $T$ in ${\bbfR}$.\par 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.
[I. M. Stancu-Minasian (Bucureşti)]
MSC 2000:
*90C32 Fractional programming
90C47 Minimax problems

Keywords: Wolfe type dual; Mond-Weir type dual; one parameter dual type; duality theorems

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