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Generalized Clarke epiderivatives of the extremum multifunction to a multi-objective parametric discrete optimal control problem
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi, Vietnam |
This paper deals with the generalized Clarke epiderivative of the extremum multifunction of a multi-objective parametric convex discrete optimal control problem with linear state equations and control constraints. By establishing an abstract result on the generalized epiderivative of the extremum multifunction of a multi-objective parametric convex mathematical programming problem, we derive a formula for computing the generalized Clarke epiderivative of the extremum multifunction to a multi-objective parametric convex discrete optimal control problem. Examples are given to illustrate the obtained results.
References:
[1] |
D. T. V. An and N. T. Toan,
Differential stability of convex discrete optimal control problem, Acta Math. Vietnam., 43 (2018), 201-217.
doi: 10.1007/s40306-017-0227-y. |
[2] |
J. -P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, In: Nachbin, L. (ed. ) Mathematical Analysis and Applications, Academic Press, New York, (1981), 159–229. |
[3] |
E. M. Bednarczuk and W. Song,
Contingent epiderivate and its applications to set-valued maps, Control Cybern., 27 (1998), 375-386.
|
[4] |
A. Bemporad and D. Mu$ \rm\tilde{n} $oz de la Pe$ \rm\tilde{n} $a,
Multiobjective model predictive control, Automatica J. IFAC, 45 (2009), 2823-2830.
doi: 10.1016/j.automatica.2009.09.032. |
[5] |
V. Bhaskar, S. K. Gupta and A. K. Ray,
Multiobjective optimization of an industrial wiped-film pet reactor, Am. Inst. Chem. Eng. J., 46 (2000), 1046-1058.
doi: 10.1002/aic.690460516. |
[6] |
V. Bhaskar, S. K. Gupta and A. K. Ray,
Applications of multiobjective optimization in chemical engineering, Rev. Chem. Eng., 16 (2000), 1-54.
doi: 10.1515/REVCE.2000.16.1.1. |
[7] |
J. Blot and N. Hayek, Infinite-Horizon Optimal Control in the Discrete-Time Framework, Springer, New York, 2014.
doi: 10.1007/978-1-4614-9038-8. |
[8] |
L. Chen,
Generalized tangent epiderivative and applications to set-valued map optimization, J. Nonlinear Convex Anal., 3 (2002), 303-313.
|
[9] |
G. Y. Chen and J. Jahn,
Optimality conditions for set-valued optimization problems, Math. Meth. Oper. Res., 48 (1998), 187-200.
doi: 10.1007/s001860050021. |
[10] |
N. H. Chieu and J.-C. Yao,
Subgradients of the optimal value function in a parametric discrete optimal control problem, J. Ind. Manag. Optim., 6 (2010), 401-410.
doi: 10.3934/jimo.2010.6.401. |
[11] |
T. D. Chuong and J.-C. Yao,
Generalized Clarke epiderivatives of parametric vector optimization problems, J. Optim. Theory Appl., 146 (2010), 77-94.
doi: 10.1007/s10957-010-9646-9. |
[12] |
F. H. Clarke, Method of Dynamic and Nonsmooth Optimization, SIAM, Philadelphia, 1989.
doi: 10.1137/1.9781611970142. |
[13] |
F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990.
doi: 10.1137/1.9781611971309. |
[14] |
E. J. Dockner, S. Jorgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511805127.![]() ![]() ![]() |
[15] |
E. Dockner and N. V. Long,
International pollution control: Cooperative versus non-cooperative strategies, J. Environ. Econ. Manag., 25 (1993), 13-29.
|
[16] |
E. J. Dockner and K. Nishimura,
Strategic growth, J. Differ. Equ. Appl., 10 (2004), 515-527.
doi: 10.1080/1023619042000193667. |
[17] |
N. Hayek,
Infinite horizon multiobjective optimal control problems in the discrete time case, Optimization, 60 (2011), 509-529.
doi: 10.1080/02331930903480352. |
[18] |
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. |
[19] |
J. Jahn, Vector Optimization. Theory, Applications and Extensions, Springer, Berlin, 2004.
doi: 10.1007/978-3-540-24828-6. |
[20] |
J. Jahn and R. Rauh,
Contingent epiderivatives and set-valued optimization, Math. Methods Oper. Res., 46 (1997), 193-211.
doi: 10.1007/BF01217690. |
[21] |
C. Y. Kaya and H. Maurer,
A numerical method for nonconvex multi-objective optimal control problems, Comput. Optim. Appl., 57 (2014), 685-702.
doi: 10.1007/s10589-013-9603-2. |
[22] |
H. Kuk, T. Tanino and M. Tanaka,
Sensitivity analysis in parametrized convex vector optimization, J. Math. Anal. Appl., 202 (1996), 511-522.
doi: 10.1006/jmaa.1996.0331. |
[23] |
H. Kuk, T. Tanino and M. Tanaka,
Sensitivity analysis in vector optimization, J. Optim. Theory Appl., 89 (1996), 713-730.
doi: 10.1007/BF02275356. |
[24] |
D. T. Luc, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, 1989. |
[25] |
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation I. Basis Theory, Springer, Berlin, 2006. |
[26] |
B. S. Mordukhovich and N. M. Nam,
Variational stability and marginal functions via generalized differentiation, Math. Oper. Res., 30 (2005), 800-816.
doi: 10.1287/moor.1050.0147. |
[27] |
M. Moussaoui and A. Seeger,
Sensitivity analysis of optimal value functions of convex parametric programs with possibly empty solution sets, SIAM J. Optim., 4 (1994), 659-675.
doi: 10.1137/0804038. |
[28] |
T.-N. Ngo and N. Hayek,
Necessary conditions of Pareto optimality for multiobjective optimal control problems under constraints, Optimization, 66 (2017), 149-177.
doi: 10.1080/02331934.2016.1261349. |
[29] |
S. Peitz and M. Dellnitz, A survey of recent trends in multiobjective optimal control - surrogate models, feedback control and objective reduction, Math. Comput. Appl., 23 (2018), Paper No. 30, 33 pp.
doi: 10.3390/mca23020030. |
[30] |
J.-P. Penot,
Differetiability properties of optimal value functions, Canad. J. Math., 56 (2004), 825-842.
doi: 10.4153/CJM-2004-037-x. |
[31] |
R. T. Rockafellar,
Hamilton-Jacobi theory and parametric analysis in fully convex problems of optimal control, J. Global Optim., 28 (2004), 419-431.
doi: 10.1023/B:JOGO.0000026459.51919.0e. |
[32] |
R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer, Berlin, 1998.
doi: 10.1007/978-3-642-02431-3. |
[33] |
D. S. Shi,
Contingent derivative of the perturbation map in multiobjective optimization, J. Optim. Theory Appl., 70 (1991), 385-396.
doi: 10.1007/BF00940634. |
[34] |
D. S. Shi,
Sensitivity analysis in convex vector optimization, J. Optim. Theory Appl., 77 (1993), 145-159.
doi: 10.1007/BF00940783. |
[35] |
W. Song and L.-J. Wan,
Contingent epidifferentiability of the value map in vector optimization, Heilongjiang Daxue Ziran Kexue Xuebao, 22 (2005), 198-203.
|
[36] |
G. Sorger,
A dynamic common property resource problem with amenity value and extraction costs, Int. J. Econ. Theory, 1 (2005), 3-19.
doi: 10.1111/j.1742-7363.2005.00002.x. |
[37] |
T. Tanino,
Sensitivity analysis in multiobjective optimization, J. Optim. Theory Appl., 56 (1988), 479-499.
doi: 10.1007/BF00939554. |
[38] |
T. Tanino,
Stability and sensitivity analysis in convex vector optimization, SIAM J. Control Optim., 26 (1988), 521-536.
doi: 10.1137/0326031. |
[39] |
L. Q. Thuy and N. T. Toan,
Subgradients of the value function in a parametric convex optimal control problem, J. Optim. Theory Appl., 170 (2016), 43-64.
doi: 10.1007/s10957-016-0921-2. |
[40] |
N. T. Toan, L. Q. Thuy, N. V. Tuyen and Y. -B. Xiao, On the no-gap second-order optimality conditions for a multi-objective discrete optimal control problem with mixed constraints, J. Global Optim., 2020. |
[41] |
N. T. Toan and J.-C. Yao,
Mordukhovich subgradients of the value function to a parametric discrete optimal control problem, J. Global Optim., 58 (2014), 595-612.
doi: 10.1007/s10898-013-0062-1. |
[42] |
R. Vinter, Optimal Control, Birkh$\rm\ddot{a}$user, Boston, 2000. |
[43] |
B. Vroemen and B. De Jager, Multiobjective control: An overview, Proceeding of the 36th IEEE Conference on Decision and Control, San Diego CA, (1997), 440–445.
doi: 10.1109/CDC. 1997.650664. |
[44] |
Z. Wu,
Tangent cone and contingent cone to the intersection of two closed sets, Nonlinear Anal., 73 (2010), 1203-1220.
doi: 10.1016/j.na.2010.04.042. |
[45] |
X. Q. Yang and K. L. Teo,
Necessary optimality conditions for bicriterion discrete time optimal control problems, J. Aust. Math. Soc. Ser. B., 40 (1999), 392-402.
doi: 10.1017/S0334270000010973. |
[46] |
V. M. Zavala and A. Flores-Tlacuahuac,
Stability of multiobjective predictive control: A utopia-tracking approach, Automatica J. IFAC, 48 (2012), 2627-2632.
doi: 10.1016/j.automatica.2012.06.066. |
show all references
References:
[1] |
D. T. V. An and N. T. Toan,
Differential stability of convex discrete optimal control problem, Acta Math. Vietnam., 43 (2018), 201-217.
doi: 10.1007/s40306-017-0227-y. |
[2] |
J. -P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, In: Nachbin, L. (ed. ) Mathematical Analysis and Applications, Academic Press, New York, (1981), 159–229. |
[3] |
E. M. Bednarczuk and W. Song,
Contingent epiderivate and its applications to set-valued maps, Control Cybern., 27 (1998), 375-386.
|
[4] |
A. Bemporad and D. Mu$ \rm\tilde{n} $oz de la Pe$ \rm\tilde{n} $a,
Multiobjective model predictive control, Automatica J. IFAC, 45 (2009), 2823-2830.
doi: 10.1016/j.automatica.2009.09.032. |
[5] |
V. Bhaskar, S. K. Gupta and A. K. Ray,
Multiobjective optimization of an industrial wiped-film pet reactor, Am. Inst. Chem. Eng. J., 46 (2000), 1046-1058.
doi: 10.1002/aic.690460516. |
[6] |
V. Bhaskar, S. K. Gupta and A. K. Ray,
Applications of multiobjective optimization in chemical engineering, Rev. Chem. Eng., 16 (2000), 1-54.
doi: 10.1515/REVCE.2000.16.1.1. |
[7] |
J. Blot and N. Hayek, Infinite-Horizon Optimal Control in the Discrete-Time Framework, Springer, New York, 2014.
doi: 10.1007/978-1-4614-9038-8. |
[8] |
L. Chen,
Generalized tangent epiderivative and applications to set-valued map optimization, J. Nonlinear Convex Anal., 3 (2002), 303-313.
|
[9] |
G. Y. Chen and J. Jahn,
Optimality conditions for set-valued optimization problems, Math. Meth. Oper. Res., 48 (1998), 187-200.
doi: 10.1007/s001860050021. |
[10] |
N. H. Chieu and J.-C. Yao,
Subgradients of the optimal value function in a parametric discrete optimal control problem, J. Ind. Manag. Optim., 6 (2010), 401-410.
doi: 10.3934/jimo.2010.6.401. |
[11] |
T. D. Chuong and J.-C. Yao,
Generalized Clarke epiderivatives of parametric vector optimization problems, J. Optim. Theory Appl., 146 (2010), 77-94.
doi: 10.1007/s10957-010-9646-9. |
[12] |
F. H. Clarke, Method of Dynamic and Nonsmooth Optimization, SIAM, Philadelphia, 1989.
doi: 10.1137/1.9781611970142. |
[13] |
F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990.
doi: 10.1137/1.9781611971309. |
[14] |
E. J. Dockner, S. Jorgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511805127.![]() ![]() ![]() |
[15] |
E. Dockner and N. V. Long,
International pollution control: Cooperative versus non-cooperative strategies, J. Environ. Econ. Manag., 25 (1993), 13-29.
|
[16] |
E. J. Dockner and K. Nishimura,
Strategic growth, J. Differ. Equ. Appl., 10 (2004), 515-527.
doi: 10.1080/1023619042000193667. |
[17] |
N. Hayek,
Infinite horizon multiobjective optimal control problems in the discrete time case, Optimization, 60 (2011), 509-529.
doi: 10.1080/02331930903480352. |
[18] |
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. |
[19] |
J. Jahn, Vector Optimization. Theory, Applications and Extensions, Springer, Berlin, 2004.
doi: 10.1007/978-3-540-24828-6. |
[20] |
J. Jahn and R. Rauh,
Contingent epiderivatives and set-valued optimization, Math. Methods Oper. Res., 46 (1997), 193-211.
doi: 10.1007/BF01217690. |
[21] |
C. Y. Kaya and H. Maurer,
A numerical method for nonconvex multi-objective optimal control problems, Comput. Optim. Appl., 57 (2014), 685-702.
doi: 10.1007/s10589-013-9603-2. |
[22] |
H. Kuk, T. Tanino and M. Tanaka,
Sensitivity analysis in parametrized convex vector optimization, J. Math. Anal. Appl., 202 (1996), 511-522.
doi: 10.1006/jmaa.1996.0331. |
[23] |
H. Kuk, T. Tanino and M. Tanaka,
Sensitivity analysis in vector optimization, J. Optim. Theory Appl., 89 (1996), 713-730.
doi: 10.1007/BF02275356. |
[24] |
D. T. Luc, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, 1989. |
[25] |
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation I. Basis Theory, Springer, Berlin, 2006. |
[26] |
B. S. Mordukhovich and N. M. Nam,
Variational stability and marginal functions via generalized differentiation, Math. Oper. Res., 30 (2005), 800-816.
doi: 10.1287/moor.1050.0147. |
[27] |
M. Moussaoui and A. Seeger,
Sensitivity analysis of optimal value functions of convex parametric programs with possibly empty solution sets, SIAM J. Optim., 4 (1994), 659-675.
doi: 10.1137/0804038. |
[28] |
T.-N. Ngo and N. Hayek,
Necessary conditions of Pareto optimality for multiobjective optimal control problems under constraints, Optimization, 66 (2017), 149-177.
doi: 10.1080/02331934.2016.1261349. |
[29] |
S. Peitz and M. Dellnitz, A survey of recent trends in multiobjective optimal control - surrogate models, feedback control and objective reduction, Math. Comput. Appl., 23 (2018), Paper No. 30, 33 pp.
doi: 10.3390/mca23020030. |
[30] |
J.-P. Penot,
Differetiability properties of optimal value functions, Canad. J. Math., 56 (2004), 825-842.
doi: 10.4153/CJM-2004-037-x. |
[31] |
R. T. Rockafellar,
Hamilton-Jacobi theory and parametric analysis in fully convex problems of optimal control, J. Global Optim., 28 (2004), 419-431.
doi: 10.1023/B:JOGO.0000026459.51919.0e. |
[32] |
R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer, Berlin, 1998.
doi: 10.1007/978-3-642-02431-3. |
[33] |
D. S. Shi,
Contingent derivative of the perturbation map in multiobjective optimization, J. Optim. Theory Appl., 70 (1991), 385-396.
doi: 10.1007/BF00940634. |
[34] |
D. S. Shi,
Sensitivity analysis in convex vector optimization, J. Optim. Theory Appl., 77 (1993), 145-159.
doi: 10.1007/BF00940783. |
[35] |
W. Song and L.-J. Wan,
Contingent epidifferentiability of the value map in vector optimization, Heilongjiang Daxue Ziran Kexue Xuebao, 22 (2005), 198-203.
|
[36] |
G. Sorger,
A dynamic common property resource problem with amenity value and extraction costs, Int. J. Econ. Theory, 1 (2005), 3-19.
doi: 10.1111/j.1742-7363.2005.00002.x. |
[37] |
T. Tanino,
Sensitivity analysis in multiobjective optimization, J. Optim. Theory Appl., 56 (1988), 479-499.
doi: 10.1007/BF00939554. |
[38] |
T. Tanino,
Stability and sensitivity analysis in convex vector optimization, SIAM J. Control Optim., 26 (1988), 521-536.
doi: 10.1137/0326031. |
[39] |
L. Q. Thuy and N. T. Toan,
Subgradients of the value function in a parametric convex optimal control problem, J. Optim. Theory Appl., 170 (2016), 43-64.
doi: 10.1007/s10957-016-0921-2. |
[40] |
N. T. Toan, L. Q. Thuy, N. V. Tuyen and Y. -B. Xiao, On the no-gap second-order optimality conditions for a multi-objective discrete optimal control problem with mixed constraints, J. Global Optim., 2020. |
[41] |
N. T. Toan and J.-C. Yao,
Mordukhovich subgradients of the value function to a parametric discrete optimal control problem, J. Global Optim., 58 (2014), 595-612.
doi: 10.1007/s10898-013-0062-1. |
[42] |
R. Vinter, Optimal Control, Birkh$\rm\ddot{a}$user, Boston, 2000. |
[43] |
B. Vroemen and B. De Jager, Multiobjective control: An overview, Proceeding of the 36th IEEE Conference on Decision and Control, San Diego CA, (1997), 440–445.
doi: 10.1109/CDC. 1997.650664. |
[44] |
Z. Wu,
Tangent cone and contingent cone to the intersection of two closed sets, Nonlinear Anal., 73 (2010), 1203-1220.
doi: 10.1016/j.na.2010.04.042. |
[45] |
X. Q. Yang and K. L. Teo,
Necessary optimality conditions for bicriterion discrete time optimal control problems, J. Aust. Math. Soc. Ser. B., 40 (1999), 392-402.
doi: 10.1017/S0334270000010973. |
[46] |
V. M. Zavala and A. Flores-Tlacuahuac,
Stability of multiobjective predictive control: A utopia-tracking approach, Automatica J. IFAC, 48 (2012), 2627-2632.
doi: 10.1016/j.automatica.2012.06.066. |
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