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Boţ, Radu Ioan; Hodrea, Ioan Bogdan; Wanka, Gert

Farkas-type results for fractional programming problems. (English) Zbl 1278.90395

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 6, 1690-1703 (2007).
Summary: Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel-Lagrange duality approach applied to an optimization problem with convex or difference of convex (DC) objective functions and finitely many convex constraints. These are obtained from the initial fractional programming problem using an idea due to Dinkelbach. We also show that our general results encompass as special cases some recently obtained Farkas-type results.

Cited in 13 Documents

MSC:

90C32 Fractional programming
49N15 Duality theory (optimization)
90C26 Nonconvex programming, global optimization

Keywords:

Farkas-type results; fractional programming; DCfunctions; conjugate functions; conjugate duality
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References:

[1] Bector, C. R., Programming Problems with Convex Fractional Functions (1967), Indian Institute of Technology, pp. 383-391 · Zbl 0159.48505
[2] Bector, C. R., Duality in nonlinear fractional programming, Zeitschrift für Operations Research, 17, 183-193 (1973) · Zbl 0267.90086
[3] R.I. Boţ, I.B. Hodrea, G. Wanka, Some new Farkas-Type results for inequality systems with D.C. functions, Chemnitz University of Technology, 2005 (Preprint 2005-4); R.I. Boţ, I.B. Hodrea, G. Wanka, Some new Farkas-Type results for inequality systems with D.C. functions, Chemnitz University of Technology, 2005 (Preprint 2005-4)
[4] Boţ, R. I.; Hodrea, I. B.; Wanka, G., Farkas-type results for inequality systems with composed convex functions via conjugate duality, Journal of Mathematical Analysis and Applications, 322, 1, 316-328 (2005) · Zbl 1104.90054
[5] Boţ, R. I.; Kassay, G.; Wanka, G., Strong duality for generalized convex optimization problems, Journal of Optimization Theory and Applications, 127, 1, 45-70 (2005) · Zbl 1158.90420
[6] Boţ, R. I.; Wanka, G., Farkas-type results with conjugate functions, SIAM Journal on Optimization, 15, 2, 540-554 (2005) · Zbl 1114.90147
[7] Boţ, R. I.; Wanka, G., Duality for multiobjective optimization problems with convex objective functions and D.C. constraints, Journal of Mathematical Analysis and Applications, 315, 2, 526-543 (2005) · Zbl 1156.90015
[8] Craven, B. D.; Mond, B., The dual of a fractional linear program, Journal of Mathematical Analysis and Applications, 42, 507-512 (1973) · Zbl 0259.90048
[9] Dinkelbach, W., On nonlinear fractional programming, Management Science, 13, 7, 492-498 (1967) · Zbl 0152.18402
[10] Jeyakumar, V., Characterizing set containments involving infinite convex constraints and reverse-convex constraints, SIAM Journal of Optimization, 13, 4, 947-959 (2003) · Zbl 1038.90061
[11] Lin, J. Y.; Sheu, R. L., Modified Dinkelbach-type algorithm for generalized fractional programs with infinitely many ratios, Journal of Optimization Theory and Applications, 126, 2, 323-343 (2005) · Zbl 1129.90053
[12] Martínez-Legaz, J. E.; Volle, M., Duality in D. C. programming: The case of several D. C. constraints, Journal of Mathematical Analysis and Applications, 237, 657-671 (1999) · Zbl 0946.90064
[13] Rockafellar, R. T., Convex Analysis (1970), Princeton University Press: Princeton University Press Princeton · Zbl 0229.90020
[14] Schaible, S., Duality in fractional programming: A unified approach, Operations Research, 24, 452-461 (1976) · Zbl 0348.90120
[15] Wanka, G.; Boţ, R. I., On the relations between different dual problems in convex mathematical programming, (Chamoni, P.; Leisten, R.; Martin, A.; Minnermann, J.; Stadtler, H., Operations Research Proceedings 2001 (2002), Springer Verlag: Springer Verlag Berlin), 255-262 · Zbl 1146.90494
[16] Yang, X. M.; Hou, S. H., On minimax fractional optimality and duality with generalized convexity, Journal of Global Optimization, 31, 2, 235-252 (2005) · Zbl 1090.90187
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