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A global optimization approach to fractional optimal control
1. | Institute of Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia |
2. | Department of Mathematics and Statistics, Curtin University, Perth, Western Australia, WA 6845, Australia, Australia |
References:
[1] |
N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific, Singapore, 2006.
doi: 10.1142/6262. |
[2] |
Y. Almogy and O. Levin, A class of fractional programming problems, Operations Research, 19 (1971), 57-67.
doi: 10.1287/opre.19.1.57. |
[3] |
C. R. Bector, Duality in nonlinear fractional programming, Zeitschrift fur Operations Research, 17 (1973), A183-A193. |
[4] |
H. P. Benson, Global optimization algorithm for the nonlinear sum of ratios problems, Journal of Optimization Theory and Applications, 112 (2002), 1-29.
doi: 10.1023/A,1013072027218. |
[5] |
I. Bykadorov, A. Ellero, S. Funari and E. Moretti, A fractional Optimal Control Problem for Maximizing Advertising Efficiency,, Working Paper n. 158/2007., ().
|
[6] |
I. Bykadorov, A. Ellero, S. Funari and E. Moretti, Dinkelbach approach to solving a class of fractional optimal control problems, Journal of Optimization Theory & Applications, 142 (2009), 55-66.
doi: 10.1007/s10957-009-9540-5. |
[7] |
A. Cambini, E. Castagnoli, L. Martein, P. Mazzoleni and S. Schaible, Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems, 345, Springer, Berlin, 1990.
doi: 10.1007/978-3-642-46709-7. |
[8] |
B. D. Craven, fractional Programming, Sigma Series in Aplied Mathematics, 4, Heldermann Verlag, 1988. |
[9] |
J. P. Crouzeix, J. A. Ferland and S. Schaible, Duality in generalized linear fractional programming, Mathematical Programming, 27 (1983), 342-354.
doi: 10.1007/BF02591908. |
[10] |
W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498.
doi: 10.1287/mnsc.13.7.492. |
[11] |
W. K. Donald, The Walrasion Vision of the Microeconomy, The university of Michigan Press, 1994. |
[12] |
R. Enkhbat, Quasiconvex progarmming, Lambert Publisher, 2009. |
[13] |
R. Enkhbat and T. Ibaraki, On the maximization and minimization of quasiconvex function, International Journal of Nonlinear and Convex Analysis, 4 (2003), 43-76. |
[14] |
N. Hadjisavvas, J. E. Martinez-Legaz and J. P. Penot, Generalized Convexity and Generalized Monotonicity, Lecture Notes in Economics and Mathematical Systems, 502, Springer, Berlin, 2001.
doi: 10.1007/978-3-642-56645-5. |
[15] |
R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Springer, Berlin, 1993.
doi: 10.1007/978-3-662-02947-3. |
[16] |
T. Ibaraki, Parametric approaches to fractional programs, Mathematical Programming, 26 (1983), 345-362.
doi: 10.1007/BF02591871. |
[17] |
L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER3 Optimal Control Software, Theory and User Manual, University of Western Australia, Perth, Australia, 1990. |
[18] |
B. Kheirfam, Multi-parametric sensitivity analysis of the constraint matrix in piecewise linear fractional programming, Journal of Industrial and Management Optimization, 6 (2010), 347-361.
doi: 10.3934/jimo.2010.6.347. |
[19] |
H. Konno and T. Kuno, Generalized linear multiplicative and fractional programming, Annals of Operations Research, 25 (1990), 147-161.
doi: 10.1007/BF02283691. |
[20] |
Lo and C. MacKinlay, Maximizing predictability in the stock and bond markets, Macroeconomic Dynamics, 1 (1997), 102-134. |
[21] |
X. J. Long and J. Quan, Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity, Journal of Industrial and Management Optimization, 1 (2011), 361-370.
doi: 10.3934/naco.2011.1.361. |
[22] |
Cs. Meszaros and T. Rapcsak, On sensitivity analysis for a class of decision systems, Decision Support Systems, 16 (1996), 231-240. |
[23] |
H. Nicolas, K. Sandar and S. Siegfried, Handbook of Generalized Convexity and Generalized Monotonicity, Springer, 2005.
doi: 10.1007/b101428. |
[24] |
S. Schaible, Fractional programming: Applications and algorithms, European Journal of Operational Research, 7 (1981), 111-120.
doi: 10.1016/0377-2217(81)90272-1. |
[25] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, 1st Edition, Longman Scientific and Technical, New York, 1991. |
[26] |
J.-F. Tsai, Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 37 (2005), 399-409.
doi: 10.1080/03052150500066737. |
[27] |
A. Zhang and S. Hayashi, Celis-Dennis-Tapia based approach to quadratic fractional programming problems with two quadratic constraints, Journal of Industrial and Management Optimization, 1 (2011), 83-98.
doi: 10.3934/naco.2011.1.83. |
show all references
References:
[1] |
N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific, Singapore, 2006.
doi: 10.1142/6262. |
[2] |
Y. Almogy and O. Levin, A class of fractional programming problems, Operations Research, 19 (1971), 57-67.
doi: 10.1287/opre.19.1.57. |
[3] |
C. R. Bector, Duality in nonlinear fractional programming, Zeitschrift fur Operations Research, 17 (1973), A183-A193. |
[4] |
H. P. Benson, Global optimization algorithm for the nonlinear sum of ratios problems, Journal of Optimization Theory and Applications, 112 (2002), 1-29.
doi: 10.1023/A,1013072027218. |
[5] |
I. Bykadorov, A. Ellero, S. Funari and E. Moretti, A fractional Optimal Control Problem for Maximizing Advertising Efficiency,, Working Paper n. 158/2007., ().
|
[6] |
I. Bykadorov, A. Ellero, S. Funari and E. Moretti, Dinkelbach approach to solving a class of fractional optimal control problems, Journal of Optimization Theory & Applications, 142 (2009), 55-66.
doi: 10.1007/s10957-009-9540-5. |
[7] |
A. Cambini, E. Castagnoli, L. Martein, P. Mazzoleni and S. Schaible, Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems, 345, Springer, Berlin, 1990.
doi: 10.1007/978-3-642-46709-7. |
[8] |
B. D. Craven, fractional Programming, Sigma Series in Aplied Mathematics, 4, Heldermann Verlag, 1988. |
[9] |
J. P. Crouzeix, J. A. Ferland and S. Schaible, Duality in generalized linear fractional programming, Mathematical Programming, 27 (1983), 342-354.
doi: 10.1007/BF02591908. |
[10] |
W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498.
doi: 10.1287/mnsc.13.7.492. |
[11] |
W. K. Donald, The Walrasion Vision of the Microeconomy, The university of Michigan Press, 1994. |
[12] |
R. Enkhbat, Quasiconvex progarmming, Lambert Publisher, 2009. |
[13] |
R. Enkhbat and T. Ibaraki, On the maximization and minimization of quasiconvex function, International Journal of Nonlinear and Convex Analysis, 4 (2003), 43-76. |
[14] |
N. Hadjisavvas, J. E. Martinez-Legaz and J. P. Penot, Generalized Convexity and Generalized Monotonicity, Lecture Notes in Economics and Mathematical Systems, 502, Springer, Berlin, 2001.
doi: 10.1007/978-3-642-56645-5. |
[15] |
R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Springer, Berlin, 1993.
doi: 10.1007/978-3-662-02947-3. |
[16] |
T. Ibaraki, Parametric approaches to fractional programs, Mathematical Programming, 26 (1983), 345-362.
doi: 10.1007/BF02591871. |
[17] |
L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER3 Optimal Control Software, Theory and User Manual, University of Western Australia, Perth, Australia, 1990. |
[18] |
B. Kheirfam, Multi-parametric sensitivity analysis of the constraint matrix in piecewise linear fractional programming, Journal of Industrial and Management Optimization, 6 (2010), 347-361.
doi: 10.3934/jimo.2010.6.347. |
[19] |
H. Konno and T. Kuno, Generalized linear multiplicative and fractional programming, Annals of Operations Research, 25 (1990), 147-161.
doi: 10.1007/BF02283691. |
[20] |
Lo and C. MacKinlay, Maximizing predictability in the stock and bond markets, Macroeconomic Dynamics, 1 (1997), 102-134. |
[21] |
X. J. Long and J. Quan, Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity, Journal of Industrial and Management Optimization, 1 (2011), 361-370.
doi: 10.3934/naco.2011.1.361. |
[22] |
Cs. Meszaros and T. Rapcsak, On sensitivity analysis for a class of decision systems, Decision Support Systems, 16 (1996), 231-240. |
[23] |
H. Nicolas, K. Sandar and S. Siegfried, Handbook of Generalized Convexity and Generalized Monotonicity, Springer, 2005.
doi: 10.1007/b101428. |
[24] |
S. Schaible, Fractional programming: Applications and algorithms, European Journal of Operational Research, 7 (1981), 111-120.
doi: 10.1016/0377-2217(81)90272-1. |
[25] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, 1st Edition, Longman Scientific and Technical, New York, 1991. |
[26] |
J.-F. Tsai, Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization, 37 (2005), 399-409.
doi: 10.1080/03052150500066737. |
[27] |
A. Zhang and S. Hayashi, Celis-Dennis-Tapia based approach to quadratic fractional programming problems with two quadratic constraints, Journal of Industrial and Management Optimization, 1 (2011), 83-98.
doi: 10.3934/naco.2011.1.83. |
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