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doi: 10.3934/jimo.2021088
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Generalized Clarke epiderivatives of the extremum multifunction to a multi-objective parametric discrete optimal control problem

Nguyen Thi Toan ,

School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi, Vietnam

* Corresponding author: Nguyen Thi Toan

Received  November 2020 Revised  February 2021 Early access April 2021

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.

Keywords: Multi-objective parametric convex discrete optimal control problem, extremum multifunction, generalized Clarke epiderivative, Clarke tangent cone, sensitivity analysis.
Mathematics Subject Classification: Primary: 49K40, 49J53; Secondary: 90C29, 93C55, 93C73.
Citation: Nguyen Thi Toan. Generalized Clarke epiderivatives of the extremum multifunction to a multi-objective parametric discrete optimal control problem. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021088
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.  Google Scholar

[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.  Google Scholar

[3]

E. M. Bednarczuk and W. Song, Contingent epiderivate and its applications to set-valued maps, Control Cybern., 27 (1998), 375-386.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

L. Chen, Generalized tangent epiderivative and applications to set-valued map optimization, J. Nonlinear Convex Anal., 3 (2002), 303-313.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

F. H. Clarke, Method of Dynamic and Nonsmooth Optimization, SIAM, Philadelphia, 1989. doi: 10.1137/1.9781611970142.  Google Scholar

[13]

F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611971309.  Google Scholar

[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.  Google Scholar
[15]

E. Dockner and N. V. Long, International pollution control: Cooperative versus non-cooperative strategies, J. Environ. Econ. Manag., 25 (1993), 13-29.   Google Scholar

[16]

E. J. Dockner and K. Nishimura, Strategic growth, J. Differ. Equ. Appl., 10 (2004), 515-527.  doi: 10.1080/1023619042000193667.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[20]

J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization, Math. Methods Oper. Res., 46 (1997), 193-211.  doi: 10.1007/BF01217690.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

D. T. Luc, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, 1989. Google Scholar

[25]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation I. Basis Theory, Springer, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J.-P. Penot, Differetiability properties of optimal value functions, Canad. J. Math., 56 (2004), 825-842.  doi: 10.4153/CJM-2004-037-x.  Google Scholar

[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.  Google Scholar

[32]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[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.  Google Scholar

[34]

D. S. Shi, Sensitivity analysis in convex vector optimization, J. Optim. Theory Appl., 77 (1993), 145-159.  doi: 10.1007/BF00940783.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[37]

T. Tanino, Sensitivity analysis in multiobjective optimization, J. Optim. Theory Appl., 56 (1988), 479-499.  doi: 10.1007/BF00939554.  Google Scholar

[38]

T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM J. Control Optim., 26 (1988), 521-536.  doi: 10.1137/0326031.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[42]

R. Vinter, Optimal Control, Birkh$\rm\ddot{a}$user, Boston, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

E. M. Bednarczuk and W. Song, Contingent epiderivate and its applications to set-valued maps, Control Cybern., 27 (1998), 375-386.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

L. Chen, Generalized tangent epiderivative and applications to set-valued map optimization, J. Nonlinear Convex Anal., 3 (2002), 303-313.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

F. H. Clarke, Method of Dynamic and Nonsmooth Optimization, SIAM, Philadelphia, 1989. doi: 10.1137/1.9781611970142.  Google Scholar

[13]

F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611971309.  Google Scholar

[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.  Google Scholar
[15]

E. Dockner and N. V. Long, International pollution control: Cooperative versus non-cooperative strategies, J. Environ. Econ. Manag., 25 (1993), 13-29.   Google Scholar

[16]

E. J. Dockner and K. Nishimura, Strategic growth, J. Differ. Equ. Appl., 10 (2004), 515-527.  doi: 10.1080/1023619042000193667.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[20]

J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization, Math. Methods Oper. Res., 46 (1997), 193-211.  doi: 10.1007/BF01217690.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

D. T. Luc, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, 1989. Google Scholar

[25]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation I. Basis Theory, Springer, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J.-P. Penot, Differetiability properties of optimal value functions, Canad. J. Math., 56 (2004), 825-842.  doi: 10.4153/CJM-2004-037-x.  Google Scholar

[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.  Google Scholar

[32]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[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.  Google Scholar

[34]

D. S. Shi, Sensitivity analysis in convex vector optimization, J. Optim. Theory Appl., 77 (1993), 145-159.  doi: 10.1007/BF00940783.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[37]

T. Tanino, Sensitivity analysis in multiobjective optimization, J. Optim. Theory Appl., 56 (1988), 479-499.  doi: 10.1007/BF00939554.  Google Scholar

[38]

T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM J. Control Optim., 26 (1988), 521-536.  doi: 10.1137/0326031.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[42]

R. Vinter, Optimal Control, Birkh$\rm\ddot{a}$user, Boston, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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