• Previous Article
    Sufficient optimality conditions and Mond-Weir duality results for a fractional multiobjective optimization problem
  • JIMO Home
  • This Issue
  • Next Article
    Parameterized S-type M-eigenvalue inclusion intervals for fourth-order partially symmetric tensors and its applications
doi: 10.3934/jimo.2021088
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

* 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.

Citation: Nguyen Thi Toan. Generalized Clarke epiderivatives of the extremum multifunction to a multi-objective parametric discrete optimal control problem. Journal of Industrial and 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.

[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. BhaskarS. 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. BhaskarS. 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. DocknerS. JorgensenN. 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. KukT. 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. KukT. 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. BhaskarS. 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. BhaskarS. 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. DocknerS. JorgensenN. 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. KukT. 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. KukT. 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.

[1]

Jianshe Yu, Honghua Bin, Zhiming Guo. Periodic solutions for discrete convex Hamiltonian systems via Clarke duality. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 939-950. doi: 10.3934/dcds.2006.15.939

[2]

Giorgio Gnecco, Andrea Bacigalupo. Convex combination of data matrices: PCA perturbation bounds for multi-objective optimal design of mechanical metafilters. Mathematical Foundations of Computing, 2021, 4 (4) : 253-269. doi: 10.3934/mfc.2021014

[3]

Nguyen Huy Chieu, Jen-Chih Yao. Subgradients of the optimal value function in a parametric discrete optimal control problem. Journal of Industrial and Management Optimization, 2010, 6 (2) : 401-410. doi: 10.3934/jimo.2010.6.401

[4]

Namsu Ahn, Soochan Kim. Optimal and heuristic algorithms for the multi-objective vehicle routing problem with drones for military surveillance operations. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1651-1663. doi: 10.3934/jimo.2021037

[5]

Liwei Zhang, Jihong Zhang, Yule Zhang. Second-order optimality conditions for cone constrained multi-objective optimization. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1041-1054. doi: 10.3934/jimo.2017089

[6]

Henri Bonnel, Ngoc Sang Pham. Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions. Journal of Industrial and Management Optimization, 2011, 7 (4) : 789-809. doi: 10.3934/jimo.2011.7.789

[7]

Behrouz Kheirfam, Kamal mirnia. Multi-parametric sensitivity analysis in piecewise linear fractional programming. Journal of Industrial and Management Optimization, 2008, 4 (2) : 343-351. doi: 10.3934/jimo.2008.4.343

[8]

Heinz Schättler, Urszula Ledzewicz. Fields of extremals and sensitivity analysis for multi-input bilinear optimal control problems. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4611-4638. doi: 10.3934/dcds.2015.35.4611

[9]

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

[10]

Kazimierz Malanowski, Helmut Maurer. Sensitivity analysis for state constrained optimal control problems. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 241-272. doi: 10.3934/dcds.1998.4.241

[11]

Behrouz Kheirfam. Multi-parametric sensitivity analysis of the constraint matrix in piecewise linear fractional programming. Journal of Industrial and Management Optimization, 2010, 6 (2) : 347-361. doi: 10.3934/jimo.2010.6.347

[12]

Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial and Management Optimization, 2021, 17 (2) : 695-709. doi: 10.3934/jimo.2019130

[13]

Zhongqiang Wu, Zongkui Xie. A multi-objective lion swarm optimization based on multi-agent. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022001

[14]

Azam Moradi, Jafar Razmi, Reza Babazadeh, Ali Sabbaghnia. An integrated Principal Component Analysis and multi-objective mathematical programming approach to agile supply chain network design under uncertainty. Journal of Industrial and Management Optimization, 2019, 15 (2) : 855-879. doi: 10.3934/jimo.2018074

[15]

Jian Xiong, Zhongbao Zhou, Ke Tian, Tianjun Liao, Jianmai Shi. A multi-objective approach for weapon selection and planning problems in dynamic environments. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1189-1211. doi: 10.3934/jimo.2016068

[16]

Dušan M. Stipanović, Claire J. Tomlin, George Leitmann. A note on monotone approximations of minimum and maximum functions and multi-objective problems. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 487-493. doi: 10.3934/naco.2011.1.487

[17]

Hamed Fazlollahtabar, Mohammad Saidi-Mehrabad. Optimizing multi-objective decision making having qualitative evaluation. Journal of Industrial and Management Optimization, 2015, 11 (3) : 747-762. doi: 10.3934/jimo.2015.11.747

[18]

Yuan-mei Xia, Xin-min Yang, Ke-quan Zhao. A combined scalarization method for multi-objective optimization problems. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2669-2683. doi: 10.3934/jimo.2020088

[19]

Xia Zhao, Jianping Dou. Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1263-1288. doi: 10.3934/jimo.2018095

[20]

V.N. Malozemov, A.V. Omelchenko. On a discrete optimal control problem with an explicit solution. Journal of Industrial and Management Optimization, 2006, 2 (1) : 55-62. doi: 10.3934/jimo.2006.2.55

2020 Impact Factor: 1.801

Article outline

[Back to Top]