A global optimization approach to fractional optimal control

Previous Article
The optimal portfolios based on a modified safety-first rule with risk-free saving
JIMO Home
This Issue
Next Article
Discount-offering and demand-rejection decisions for substitutable products with different profit levels

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

A global optimization approach to fractional optimal control

Enkhbat Rentsen ^1, , J. Zhou ^2, and K. L. Teo ^2,

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

Abstract
Related Papers

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.

Keywords: optimal control., Optimality condition, fractional programming.

Mathematics Subject Classification: Primary: 49B; Secondary: 49K, 65.

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.

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.

[1]	Ana P. Lemos-Paião, Cristiana J. Silva, Delfim F. M. Torres. A sufficient optimality condition for delayed state-linear optimal control problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2293-2313. doi: 10.3934/dcdsb.2019096
[2]	Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control and Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033
[3]	Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235
[4]	Mansoureh Alavi Hejazi, Soghra Nobakhtian. Optimality conditions for multiobjective fractional programming, via convexificators. Journal of Industrial and Management Optimization, 2020, 16 (2) : 623-631. doi: 10.3934/jimo.2018170
[5]	Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004
[6]	Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361
[7]	Xiao-Bing Li, Qi-Lin Wang, Zhi Lin. Optimality conditions and duality for minimax fractional programming problems with data uncertainty. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1133-1151. doi: 10.3934/jimo.2018089
[8]	Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial and Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1
[9]	Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control and Related Fields, 2020, 10 (2) : 379-403. doi: 10.3934/mcrf.2020002
[10]	Ram U. Verma. General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 333-339. doi: 10.3934/naco.2011.1.333
[11]	Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control and Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001
[12]	Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559
[13]	Chongyang Liu, Wenjuan Sun, Xiaopeng Yi. Optimal control of a fractional smoking system. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022071
[14]	Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101
[15]	Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial and Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[16]	Aleksandar Jović. Saddle-point type optimality criteria, duality and a new approach for solving nonsmooth fractional continuous-time programming problems. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022025
[17]	Heinz Schättler, Urszula Ledzewicz, Helmut Maurer. Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2657-2679. doi: 10.3934/dcdsb.2014.19.2657
[18]	Tone-Yau Huang, Tamaki Tanaka. Optimality and duality for complex multi-objective programming. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 121-134. doi: 10.3934/naco.2021055
[19]	Gang Li, Yinghong Xu, Zhenhua Qin. Optimality conditions for composite DC infinite programming problems. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022064
[20]	Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks and Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027

2021 Impact Factor: 1.411

Tools

Metrics

Other articles
by authors

on AIMS
Enkhbat Rentsen J. Zhou K. L. Teo
on Google Scholar
Enkhbat Rentsen J. Zhou K. L. Teo

[Back to Top]