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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. \[ v^{\ast }=\min_{x}\max_{1\leq i\leq p}\frac{ \int_{a}^{b}f^{i}( t,x(t),\dot{x}(t))\,dt}{ \int_{a}^{b}g^{i}( t,x(t),\dot{x}(t))\,dt} \] 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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[1] DOI: 10.1080/02331939508844040 · Zbl 0821.90112
[2] Hanson, J. Inf. Optim. Sci. 3 pp 25– (1982)
[3] Stancu-Minasian, Fractional programming theory, methods and applications (1997) · Zbl 0899.90155
[4] DOI: 10.1007/BF02591908 · Zbl 0526.90083
[5] Craven, Asia-Pacific J. Oper. Res. 10 pp 219– (1993)
[6] Craven, Workshop/Miniconference on Functional Analysis and Optimization (July 8–24, 1988) pp 24– (1988) · Zbl 0651.90084
[7] Chandra, J. Austral. Math. Soc. Ser. B 28 pp 170– (1986)
[8] Chandra, J. Austral. Math. Soc. Ser. A 39 pp 28– (1985)
[9] DOI: 10.1016/0022-247X(85)90357-9 · Zbl 0588.90078
[10] Bector, Asia-Pacific J. Oper Res. 5 pp 134– (1988)
[11] DOI: 10.1080/02331939408843974 · Zbl 0816.49028
[12] Bector, Util. Math. 42 pp 39– (1992)
[13] Mond, J. Austral. Math. Soc. Ser. B 31 pp 108– (1989)
[14] DOI: 10.1016/0022-247X(88)90026-1 · Zbl 0663.49005
[15] DOI: 10.1080/02331939608844228 · Zbl 0874.90184
[16] DOI: 10.1080/02331939608844188 · Zbl 0854.90122
[17] DOI: 10.1006/jmaa.1998.6204 · Zbl 0916.90251
[18] DOI: 10.1023/A:1021771011210 · Zbl 0945.90078
[19] DOI: 10.1006/jmaa.2000.6715 · Zbl 1073.90543
[20] DOI: 10.1006/jmaa.1998.6187 · Zbl 0922.90121
[21] DOI: 10.1006/jmaa.1997.5644 · Zbl 0914.90232
[22] DOI: 10.1016/0022-247X(90)90218-5 · Zbl 0718.49019
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