December  2018, 11(6): 1121-1141. doi: 10.3934/dcdss.2018064

Infinite-horizon multiobjective optimal control problems for bounded processes

CRED EA 7321 Université Panthéon-Assas Paris 2, 12, Place du Panthéon, Paris, France

* Corresponding author

Received  September 2016 Revised  September 2016 Published  June 2018

This paper studies multiobjective optimal control problems in the discrete time framework and in the infinite horizon case for bounded processes. The paper generalizes to the multiobjective case results obtained for single-objective optimal control problems in that framework. The dynamics are governed by difference equations. Necessary conditions of Pareto optimality are presented namely Pontryagin maximum principles in the strong form and in the weak form. Sufficient conditions are also provided. Other notions of Pareto optimality are defined when the infinite series do not necessarily converge and links with these unbounded cases are established.

Citation: Naïla Hayek. Infinite-horizon multiobjective optimal control problems for bounded processes. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1121-1141. doi: 10.3934/dcdss.2018064
References:
[1]

V. M. Alexeev, V. M. Tikhomirov and S. V. Fomin, Commande Optimale, French translation, Mir, Moscow, 1982.

[2]

D. C. Aliprantis and K. C. Border, Infinite Dimensional Analysis, 2nd Edition, SpringerVerlag, Berlin, 1999. doi: 10.1007/978-3-662-03961-8.

[3]

S. Bellaassali and A. Jourani, Necessary optimality conditions in multiobjective dynamic optimization, SIAM J. Control Optim, 42 (2004), 2043-2061.  doi: 10.1137/S0363012902406576.

[4]

G. Bigi, Regularity conditions in vector optimization, J. Optim. Theory Appl, 102 (1999), 83-96.  doi: 10.1023/A:1021890328184.

[5]

J. Blot, Infinite-horizon Pontryagin principles without invertibility, J. Nonlinear Convex Anal., 10 (2009), 177-189. 

[6]

J. Blot and P. Cartigny, Optimality in infinite-horizon problems under signs conditions, Journal of Optimization Theory and Applications, 106 (2000), 411-419.  doi: 10.1023/A:1004611816252.

[7]

J. Blot and H. Chebbi, Discrete time Pontryagin principles with infinite horizon, J. Math. Anal. Appl., 246 (2000), 265-279.  doi: 10.1006/jmaa.2000.6797.

[8]

J. Blot and B. Crettez, On the smoothness of optimal paths, Decisions in Economics and Finance, 27 (2004), 1-34.  doi: 10.1007/s10203-004-0042-5.

[9]

J. Blot and N. Hayek, Infinite-horizon Pontryagin principles with constraints, Communications of the Laufen colloquium on science, Ruffing, A. et al, eds, Aachen: Shaker Verlag. Berichte aus der Mathematik, 2 Laufen, 2007, 1–14.

[10]

J. Blot and N. Hayek, Infinite horizon discrete time control problems for bounded processes, Advances in Difference Equations, 2008 (2008), Article ID 654267, 14 pages. doi: 10.1155/2008/654267.

[11]

J. Blot and N. Hayek, Infinite-horizon Optimal Control in the Discrete-Time Framework, SpringerBriefs in Optimization, 2014. doi: 10.1007/978-1-4614-9038-8.

[12]

J. Blot, N. Hayek, F. Pekergin and N. Pekergin, The competition between Internet service qualities from a difference games viewpoint, International Game Theory Review, 14 (2012), 1250001 (36 pages). doi: 10.1142/S0219198912500016.

[13]

J. BlotN. HayekF. Pekergin and N. Pekergin, Pontryagin principles for bounded discrete-time processes, Optimization, 64 (2015), 505-520.  doi: 10.1080/02331934.2013.766991.

[14]

V. G. Boltyanski, Commande Optimale Des Systèmes Discrets, French edition, MIR, Moscow, 1976.

[15]

D. A. Carlson, A. B. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control, Deterministic and Stochastic Systems, 2nd edition, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-76755-5.

[16]

A. Chinchuluun and P. M. Pardalos, A survey of recent developments in multiobjective optimization, Ann. Oper. Res., 154 (2007), 29-50.  doi: 10.1007/s10479-007-0186-0.

[17]

F. Gianessi, G. Mastroeni and L. Pellegrini, On the theory of vector optimization and variational inequalities. Image space analysis and separation, in Vector Variational Inequalities and Vector Equilibria, (ed F. Gianessi), Kluwer, Dordrecht, 38 (2000), 153–215. doi: 10.1007/978-1-4613-0299-5_11.

[18]

N. Hayek, Infinite horizon multiobjective optimal control problem in the discrete time case, Optimization, 60 (2011), 509-529.  doi: 10.1080/02331930903480352.

[19]

N. Hayek, A generalization of mixed problems with an application to multiobjective optimal control, J. Optim. Theory Appl., 150 (2011), 498-515.  doi: 10.1007/s10957-011-9850-2.

[20]

A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, English edition, NorthHolland Publishing Company, Amsterdam, 1979.

[21]

Y. Ishizuka and K. Shimizu, Necessary and sufficient conditions for the efficient solutions of nondifferentiable multiobjective problems, IEEE Transactions on Systems, Man and Cybernetics SMC-14, 14 (1984), 625-629.  doi: 10.1109/TSMC.1984.6313334.

[22]

J. Jahn, Vector Optimization. Theory, Applications and Extensions, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-24828-6.

[23]

V. Jeyakumar, Farkas' Lemma and Extensions, in Encyclopedia of Optimization, (eds. C. A. Floudas, P. M. Pardalos), Kluwer Academic Publishers, Boston, 2001.

[24]

B. Jimenez and V. Novo, Alternative theorems and necessary optimality conditions for directionally differentiable multiobjective programs, J. Convex Anal., 9 (2002), 97-116. 

[25]

P. Q. Khanh and T. H. Nuong, On necessary optimality conditions in vector optimization problems, J. Optim. Theory Appl., 58 (1988), 63-81.  doi: 10.1007/BF00939770.

[26]

H. W. Kuhn and A. W. Tucker, Nonlinear programming, in Proceedings of the second Berkeley symposium on mathematical statistics and probability, (ed J. Neyman), University of California Press, Berkeley and Los Angeles, (1951), 481–492.

[27]

D. Li, On general multiple linear quadratic control problems, IEEE Transactions on Automatic Control, 38 (1993), 1722-1727.  doi: 10.1109/9.262049.

[28]

J. G. Lin, Maximal vectors and multi-objective optimization, J. Optim. Theory Appl., 18 (1976), 41-64.  doi: 10.1007/BF00933793.

[29]

P. Michel, Programmes mathématiques mixtes. Application au principe du maximum en temps discret dans le cas déterministe et dans le cas stochastique, RAIRO Rech. Opér., 14 (1980), 1-19.  doi: 10.1051/ro/1980140100011.

[30]

M. Minami, Weak Pareto-optimal necessary conditions in a nondifferentiable multiobjective program on a Banach space, J. Optim. Theory Appl., 41 (1983), 451-461.  doi: 10.1007/BF00935364.

[31]

L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mitchenko, Théorie Mathématique Des Processus Optimaux, French edition. Mir, Moscow, 1974.

[32]

M. Salukvadze, On the existence of solutions in problems of optimization under vectorvalued criteria, J. Optim. Theory Appl., 13 (1974), 203-217.  doi: 10.1007/BF00935540.

[33]

L. B. SantosA. J. V. BrandaoR. Osuna-Gomez and M. A. Rojas-Medar, Necessary and sufficient conditions for weak efficiency in nonsmooth vectorial optimization problems, Optimization, 58 (2009), 981-993.  doi: 10.1080/02331930701763645.

[34]

Y, Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization, Academic Press, New York, 1985.

[35]

A. Seierstad and K. Sydsaeter, Sufficient conditions in optimal control theory, Internat. Econom. Rev., 18 (1977), 367-391.  doi: 10.2307/2525753.

[36]

C. Singh, Optimality conditions in multiobjective differentiable programming, J. Optim. Theory Appl., 53 (1987), 115-123.  doi: 10.1007/BF00938820.

[37]

P. L. Yu and G. Leitmann, Nondominated decision and cone convexity in dynamic multicriteria decision problems, J. Optim. Theory Appl., 14 (1974), 573-584.  doi: 10.1007/BF00932849.

[38]

L. A. Zadeh, Optimality and non-scalar-valued performance criteria, IEEE Transactions on Automatic Control, 8 (1963), 59-60.  doi: 10.1109/TAC.1963.1105511.

[39]

A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Nonconvex optimization and its applications, 80, Springer, New York, 2006.

[40]

A. J. Zaslavski, Turnpike Phenomenon and Infinite Horizon Optimal Control, Springer Optimization and Its Applications, 99, Springer International Publishing, 2014. doi: 10.1007/978-3-319-08828-0.

[41]

A. J. Zaslavski, Stability of the Turnpike Phenomenon for Discrete-Time Optimal Control Problems, Series Title SpringerBriefs in Optimization, Springer International Publishing, 2014. doi: 10.1007/978-3-319-08034-5.

[42]

J. Zhu, Hamiltonian necessary conditions for a multiobjective optimal control problem with endpoint constraints, SIAM J. Control Optim., 39 (2000), 97-112.  doi: 10.1137/S0363012999350821.

show all references

References:
[1]

V. M. Alexeev, V. M. Tikhomirov and S. V. Fomin, Commande Optimale, French translation, Mir, Moscow, 1982.

[2]

D. C. Aliprantis and K. C. Border, Infinite Dimensional Analysis, 2nd Edition, SpringerVerlag, Berlin, 1999. doi: 10.1007/978-3-662-03961-8.

[3]

S. Bellaassali and A. Jourani, Necessary optimality conditions in multiobjective dynamic optimization, SIAM J. Control Optim, 42 (2004), 2043-2061.  doi: 10.1137/S0363012902406576.

[4]

G. Bigi, Regularity conditions in vector optimization, J. Optim. Theory Appl, 102 (1999), 83-96.  doi: 10.1023/A:1021890328184.

[5]

J. Blot, Infinite-horizon Pontryagin principles without invertibility, J. Nonlinear Convex Anal., 10 (2009), 177-189. 

[6]

J. Blot and P. Cartigny, Optimality in infinite-horizon problems under signs conditions, Journal of Optimization Theory and Applications, 106 (2000), 411-419.  doi: 10.1023/A:1004611816252.

[7]

J. Blot and H. Chebbi, Discrete time Pontryagin principles with infinite horizon, J. Math. Anal. Appl., 246 (2000), 265-279.  doi: 10.1006/jmaa.2000.6797.

[8]

J. Blot and B. Crettez, On the smoothness of optimal paths, Decisions in Economics and Finance, 27 (2004), 1-34.  doi: 10.1007/s10203-004-0042-5.

[9]

J. Blot and N. Hayek, Infinite-horizon Pontryagin principles with constraints, Communications of the Laufen colloquium on science, Ruffing, A. et al, eds, Aachen: Shaker Verlag. Berichte aus der Mathematik, 2 Laufen, 2007, 1–14.

[10]

J. Blot and N. Hayek, Infinite horizon discrete time control problems for bounded processes, Advances in Difference Equations, 2008 (2008), Article ID 654267, 14 pages. doi: 10.1155/2008/654267.

[11]

J. Blot and N. Hayek, Infinite-horizon Optimal Control in the Discrete-Time Framework, SpringerBriefs in Optimization, 2014. doi: 10.1007/978-1-4614-9038-8.

[12]

J. Blot, N. Hayek, F. Pekergin and N. Pekergin, The competition between Internet service qualities from a difference games viewpoint, International Game Theory Review, 14 (2012), 1250001 (36 pages). doi: 10.1142/S0219198912500016.

[13]

J. BlotN. HayekF. Pekergin and N. Pekergin, Pontryagin principles for bounded discrete-time processes, Optimization, 64 (2015), 505-520.  doi: 10.1080/02331934.2013.766991.

[14]

V. G. Boltyanski, Commande Optimale Des Systèmes Discrets, French edition, MIR, Moscow, 1976.

[15]

D. A. Carlson, A. B. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control, Deterministic and Stochastic Systems, 2nd edition, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-76755-5.

[16]

A. Chinchuluun and P. M. Pardalos, A survey of recent developments in multiobjective optimization, Ann. Oper. Res., 154 (2007), 29-50.  doi: 10.1007/s10479-007-0186-0.

[17]

F. Gianessi, G. Mastroeni and L. Pellegrini, On the theory of vector optimization and variational inequalities. Image space analysis and separation, in Vector Variational Inequalities and Vector Equilibria, (ed F. Gianessi), Kluwer, Dordrecht, 38 (2000), 153–215. doi: 10.1007/978-1-4613-0299-5_11.

[18]

N. Hayek, Infinite horizon multiobjective optimal control problem in the discrete time case, Optimization, 60 (2011), 509-529.  doi: 10.1080/02331930903480352.

[19]

N. Hayek, A generalization of mixed problems with an application to multiobjective optimal control, J. Optim. Theory Appl., 150 (2011), 498-515.  doi: 10.1007/s10957-011-9850-2.

[20]

A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, English edition, NorthHolland Publishing Company, Amsterdam, 1979.

[21]

Y. Ishizuka and K. Shimizu, Necessary and sufficient conditions for the efficient solutions of nondifferentiable multiobjective problems, IEEE Transactions on Systems, Man and Cybernetics SMC-14, 14 (1984), 625-629.  doi: 10.1109/TSMC.1984.6313334.

[22]

J. Jahn, Vector Optimization. Theory, Applications and Extensions, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-24828-6.

[23]

V. Jeyakumar, Farkas' Lemma and Extensions, in Encyclopedia of Optimization, (eds. C. A. Floudas, P. M. Pardalos), Kluwer Academic Publishers, Boston, 2001.

[24]

B. Jimenez and V. Novo, Alternative theorems and necessary optimality conditions for directionally differentiable multiobjective programs, J. Convex Anal., 9 (2002), 97-116. 

[25]

P. Q. Khanh and T. H. Nuong, On necessary optimality conditions in vector optimization problems, J. Optim. Theory Appl., 58 (1988), 63-81.  doi: 10.1007/BF00939770.

[26]

H. W. Kuhn and A. W. Tucker, Nonlinear programming, in Proceedings of the second Berkeley symposium on mathematical statistics and probability, (ed J. Neyman), University of California Press, Berkeley and Los Angeles, (1951), 481–492.

[27]

D. Li, On general multiple linear quadratic control problems, IEEE Transactions on Automatic Control, 38 (1993), 1722-1727.  doi: 10.1109/9.262049.

[28]

J. G. Lin, Maximal vectors and multi-objective optimization, J. Optim. Theory Appl., 18 (1976), 41-64.  doi: 10.1007/BF00933793.

[29]

P. Michel, Programmes mathématiques mixtes. Application au principe du maximum en temps discret dans le cas déterministe et dans le cas stochastique, RAIRO Rech. Opér., 14 (1980), 1-19.  doi: 10.1051/ro/1980140100011.

[30]

M. Minami, Weak Pareto-optimal necessary conditions in a nondifferentiable multiobjective program on a Banach space, J. Optim. Theory Appl., 41 (1983), 451-461.  doi: 10.1007/BF00935364.

[31]

L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mitchenko, Théorie Mathématique Des Processus Optimaux, French edition. Mir, Moscow, 1974.

[32]

M. Salukvadze, On the existence of solutions in problems of optimization under vectorvalued criteria, J. Optim. Theory Appl., 13 (1974), 203-217.  doi: 10.1007/BF00935540.

[33]

L. B. SantosA. J. V. BrandaoR. Osuna-Gomez and M. A. Rojas-Medar, Necessary and sufficient conditions for weak efficiency in nonsmooth vectorial optimization problems, Optimization, 58 (2009), 981-993.  doi: 10.1080/02331930701763645.

[34]

Y, Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization, Academic Press, New York, 1985.

[35]

A. Seierstad and K. Sydsaeter, Sufficient conditions in optimal control theory, Internat. Econom. Rev., 18 (1977), 367-391.  doi: 10.2307/2525753.

[36]

C. Singh, Optimality conditions in multiobjective differentiable programming, J. Optim. Theory Appl., 53 (1987), 115-123.  doi: 10.1007/BF00938820.

[37]

P. L. Yu and G. Leitmann, Nondominated decision and cone convexity in dynamic multicriteria decision problems, J. Optim. Theory Appl., 14 (1974), 573-584.  doi: 10.1007/BF00932849.

[38]

L. A. Zadeh, Optimality and non-scalar-valued performance criteria, IEEE Transactions on Automatic Control, 8 (1963), 59-60.  doi: 10.1109/TAC.1963.1105511.

[39]

A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Nonconvex optimization and its applications, 80, Springer, New York, 2006.

[40]

A. J. Zaslavski, Turnpike Phenomenon and Infinite Horizon Optimal Control, Springer Optimization and Its Applications, 99, Springer International Publishing, 2014. doi: 10.1007/978-3-319-08828-0.

[41]

A. J. Zaslavski, Stability of the Turnpike Phenomenon for Discrete-Time Optimal Control Problems, Series Title SpringerBriefs in Optimization, Springer International Publishing, 2014. doi: 10.1007/978-3-319-08034-5.

[42]

J. Zhu, Hamiltonian necessary conditions for a multiobjective optimal control problem with endpoint constraints, SIAM J. Control Optim., 39 (2000), 97-112.  doi: 10.1137/S0363012999350821.

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