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January  2016, 12(1): 73-82. doi: 10.3934/jimo.2016.12.73

## 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

Received  February 2014 Revised  November 2014 Published  April 2015

In this paper, we consider a fractional optimal control problem governed by system of linear differential equations, where its cost function is expressed as the ratio of convex and concave functions. The problem is a hard nonconvex optimal control problem and application of Pontriyagin's principle does not always guarantee finding a global optimal control. Even this type of problems in a finite dimensional space is known as NP hard. This optimal control problem can, in principle, be solved by Dinkhelbach algorithm [10]. However, it leads to solving a sequence of hard D.C programming problems in its finite dimensional analogy. To overcome this difficulty, we introduce a reachable set for the linear system. In this way, the problem is reduced to a quasiconvex maximization problem in a finite dimensional space. Based on a global optimality condition, we propose an algorithm for solving this fractional optimal control problem and we show that the algorithm generates a sequence of local optimal controls with improved cost values. The proposed algorithm is then applied to several test problems, where the global optimal cost value is obtained for each case.
Citation: Enkhbat Rentsen, J. Zhou, K. L. Teo. A global optimization approach to fractional optimal control. Journal of Industrial and Management Optimization, 2016, 12 (1) : 73-82. doi: 10.3934/jimo.2016.12.73
##### 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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##### 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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