American Institute of Mathematical Sciences

Advanced
October  2011, 7(4): 789-809. doi: 10.3934/jimo.2011.7.789

Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions

Henri Bonnel 1, and  Ngoc Sang Pham 1,

1. 

Equipe de Recherhce en Informatique et Mathématiques (ERIM), University of New Caledonia (France), B.P. R4, F98851, Nouméa Cedex, New Caledonia (French), New Caledonia (French)

Received  October 2010 Revised  May 2011 Published  August 2011

We present explicit optimality conditions for a nonsmooth functional defined over the (properly or weakly) Pareto set associated with a multi-objective linear-quadratic control problem. This problem is very difficult even in a finite dimensional setting , i.e. when, instead of a control problem, we deal with a mathematical programming problem. Amongst various applications, our problem may be considered as a response for a decision maker when he has to choose a solution over the solution set of the grand coalition $p$-player cooperative differential game.
Keywords: Grand coalition $p$-player cooperative differential game, Optimizing over the Pareto set., Multi-objective Control and Optimization.
Mathematics Subject Classification: Primary: 90C29, 91A23, 49K30; Secondary: 49N10, 91A1.
Citation: 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 & Management Optimization, 2011, 7 (4) : 789-809. doi: 10.3934/jimo.2011.7.789
References:
[1]

H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, "Matrix Riccati Equations in Control and Systems Theory,", Systems & Control: Foundations & Applications, (2003).  doi: 10.1007/978-3-0348-8081-7_9.  Google Scholar

[2]

L. T. H. An, P. D. Tao and L. D. Muu, Numerical solution for optimization over the efficient set by d.c. optimization algorithms,, Oper. Res. Lett., 19 (1996), 117.  doi: 10.1016/0167-6377(96)00022-3.  Google Scholar

[3]

J.-P. Aubin and I. Ekeland, "Applied Nonlinear Analysis,", Pure and Applied Mathematics (New York), (1984).   Google Scholar

[4]

V. Azhmyakov, An approach to controlled mechanical systems based on the multiobjective optimization technique,, J. Ind. Manag. Optim., 4 (2008), 697.   Google Scholar

[5]

H. P. Benson, Optimization over the efficient set,, J. Math. Anal. Appl., 98 (1984), 562.  doi: 10.1016/0022-247X(84)90269-5.  Google Scholar

[6]

H. P. Benson, A finite, nonadjacent extreme point search algorithm for optimization over the efficient set,, J. Optim. Theory Appl., 73 (1992), 47.  doi: 10.1007/BF00940077.  Google Scholar

[7]

S. Bolintineanu, Optimality conditions for minimization over the (weakly or properly) efficient set,, J. Math. Anal. Appl., 173 (1993), 523.   Google Scholar

[8]

S. Bolintineanu, Minimization of a quasi-concave function over an efficient set,, Math. Programming, 61 (1993), 89.  doi: 10.1007/BF01582141.  Google Scholar

[9]

S. Bolintineanu, Necessary conditions for nonlinear suboptimization over the weakly-efficient set,, J. Optim. Theory Appl., 78 (1993), 579.  doi: 10.1007/BF00939883.  Google Scholar

[10]

S. Bolintinéanu and M. El Maghri, Pénalisation dans l'optimisation sur l'ensemble faiblement efficient,, (French) [Penalization in optimization over the weakly efficient set], 31 (1997), 295.   Google Scholar

[11]

H. Bonnel and C. Y. Kaya, Optimization over the efficient set of multi-objective convex optimal control problems,, J. Optim. Theory Appl., 147 (2010), 93.  doi: 10.1007/s10957-010-9709-y.  Google Scholar

[12]

H. Bonnel and J. Morgan, Semivectorial bilevel optimization problem: Penalty approach,, J. Optim. Theory Appl., 131 (2006), 365.  doi: 10.1007/s10957-006-9150-4.  Google Scholar

[13]

H. Bonnel, Optimality conditions for the semivectorial bilevel optimization problem,, Pac. J. Optim., 2 (2006), 447.   Google Scholar

[14]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", Canadian Mathematical Society Series of Monographs and Advanced Texts, (1983).   Google Scholar

[15]

B. D. Craven, Aspects of multicriteria optimization,, in, (1991), 93.   Google Scholar

[16]

J. P. Dauer, Optimization over the efficient set using an active constraint approach,, Z. Oper. Res., 35 (1991), 185.   Google Scholar

[17]

J. P. Dauer and T. A. Fosnaugh, Optimization over the efficient set,, J. Global Optim., 7 (1995), 261.  doi: 10.1007/BF01279451.  Google Scholar

[18]

G. Eichfelder, "Adaptive Scalarization Methods in Multiobjective Optimization,", Springer-Verlag, (2008).  doi: 10.1007/978-3-540-79159-1.  Google Scholar

[19]

J. Engwerda, Necessary and sufficient conditions for Pareto optimal solution of cooperative differential games,, SIAM J. Control Optim., 48 (2010), 3859.  doi: 10.1137/080726227.  Google Scholar

[20]

J. Fülöp, A cutting plane algorithm for linear optimization over the efficient set,, in, 405 (1994), 374.   Google Scholar

[21]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces,", CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 17 (2003).   Google Scholar

[22]

R. Horst and N. V. Thoai, Maximizing a concave function over the efficient or weakly-efficient set,, European J. Oper. Res., 117 (1999), 239.   Google Scholar

[23]

R. Horst, N. V. Thoai, Y. Yamamoto and D. Zenke, On optimization over the efficient set in linear multicriteria programming,, J. Optim. Theory Appl., 134 (2007), 433.  doi: 10.1007/s10957-007-9219-8.  Google Scholar

[24]

J. Jahn, "Vector Optimization: Theory, Applications, and Extensions,", Springer-Verlag, (2004).   Google Scholar

[25]

J. Jahn, "Introduction to the Theory of Nonlinear Optimization,", 3rd edition, (2007).   Google Scholar

[26]

Y. Liu, K. L. Teo and R. P. Agarwal, A general approach to nonlinear multiple control problems with perturbation consideration,, Math. Comput. Modelling, 26 (1997), 49.  doi: 10.1016/S0895-7177(97)00239-2.  Google Scholar

[27]

D. T. Lųc, "Theory of Vector Optimization,", Lecture Notes in Economics and Mathematical Systems, 319 (1989).   Google Scholar

[28]

K. Miettinen, "Nonlinear Multiobjective Optimization,", International Series in Operations Research & Management Science, 12 (1999).  doi: 10.1007/978-1-4615-5563-6.  Google Scholar

[29]

J. Philip, Algorithms for the vector maximization problem,, Math. Programming, 2 (1972), 207.  doi: 10.1007/BF01584543.  Google Scholar

[30]

T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar

[31]

K. L. Teo, D. Li and Y. Liu, Perturbation feedback control in general multiple linear-quadratic control problems,, IMA J. Math. Control Inform., 15 (1998), 303.  doi: 10.1093/imamci/15.3.303.  Google Scholar

[32]

Y. Yamamoto, Optimization over the efficient set: Overview,, J. Global Optim., 22 (2002), 285.  doi: 10.1023/A:1013875600711.  Google Scholar

show all references

References:
[1]

H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, "Matrix Riccati Equations in Control and Systems Theory,", Systems & Control: Foundations & Applications, (2003).  doi: 10.1007/978-3-0348-8081-7_9.  Google Scholar

[2]

L. T. H. An, P. D. Tao and L. D. Muu, Numerical solution for optimization over the efficient set by d.c. optimization algorithms,, Oper. Res. Lett., 19 (1996), 117.  doi: 10.1016/0167-6377(96)00022-3.  Google Scholar

[3]

J.-P. Aubin and I. Ekeland, "Applied Nonlinear Analysis,", Pure and Applied Mathematics (New York), (1984).   Google Scholar

[4]

V. Azhmyakov, An approach to controlled mechanical systems based on the multiobjective optimization technique,, J. Ind. Manag. Optim., 4 (2008), 697.   Google Scholar

[5]

H. P. Benson, Optimization over the efficient set,, J. Math. Anal. Appl., 98 (1984), 562.  doi: 10.1016/0022-247X(84)90269-5.  Google Scholar

[6]

H. P. Benson, A finite, nonadjacent extreme point search algorithm for optimization over the efficient set,, J. Optim. Theory Appl., 73 (1992), 47.  doi: 10.1007/BF00940077.  Google Scholar

[7]

S. Bolintineanu, Optimality conditions for minimization over the (weakly or properly) efficient set,, J. Math. Anal. Appl., 173 (1993), 523.   Google Scholar

[8]

S. Bolintineanu, Minimization of a quasi-concave function over an efficient set,, Math. Programming, 61 (1993), 89.  doi: 10.1007/BF01582141.  Google Scholar

[9]

S. Bolintineanu, Necessary conditions for nonlinear suboptimization over the weakly-efficient set,, J. Optim. Theory Appl., 78 (1993), 579.  doi: 10.1007/BF00939883.  Google Scholar

[10]

S. Bolintinéanu and M. El Maghri, Pénalisation dans l'optimisation sur l'ensemble faiblement efficient,, (French) [Penalization in optimization over the weakly efficient set], 31 (1997), 295.   Google Scholar

[11]

H. Bonnel and C. Y. Kaya, Optimization over the efficient set of multi-objective convex optimal control problems,, J. Optim. Theory Appl., 147 (2010), 93.  doi: 10.1007/s10957-010-9709-y.  Google Scholar

[12]

H. Bonnel and J. Morgan, Semivectorial bilevel optimization problem: Penalty approach,, J. Optim. Theory Appl., 131 (2006), 365.  doi: 10.1007/s10957-006-9150-4.  Google Scholar

[13]

H. Bonnel, Optimality conditions for the semivectorial bilevel optimization problem,, Pac. J. Optim., 2 (2006), 447.   Google Scholar

[14]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", Canadian Mathematical Society Series of Monographs and Advanced Texts, (1983).   Google Scholar

[15]

B. D. Craven, Aspects of multicriteria optimization,, in, (1991), 93.   Google Scholar

[16]

J. P. Dauer, Optimization over the efficient set using an active constraint approach,, Z. Oper. Res., 35 (1991), 185.   Google Scholar

[17]

J. P. Dauer and T. A. Fosnaugh, Optimization over the efficient set,, J. Global Optim., 7 (1995), 261.  doi: 10.1007/BF01279451.  Google Scholar

[18]

G. Eichfelder, "Adaptive Scalarization Methods in Multiobjective Optimization,", Springer-Verlag, (2008).  doi: 10.1007/978-3-540-79159-1.  Google Scholar

[19]

J. Engwerda, Necessary and sufficient conditions for Pareto optimal solution of cooperative differential games,, SIAM J. Control Optim., 48 (2010), 3859.  doi: 10.1137/080726227.  Google Scholar

[20]

J. Fülöp, A cutting plane algorithm for linear optimization over the efficient set,, in, 405 (1994), 374.   Google Scholar

[21]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, "Variational Methods in Partially Ordered Spaces,", CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 17 (2003).   Google Scholar

[22]

R. Horst and N. V. Thoai, Maximizing a concave function over the efficient or weakly-efficient set,, European J. Oper. Res., 117 (1999), 239.   Google Scholar

[23]

R. Horst, N. V. Thoai, Y. Yamamoto and D. Zenke, On optimization over the efficient set in linear multicriteria programming,, J. Optim. Theory Appl., 134 (2007), 433.  doi: 10.1007/s10957-007-9219-8.  Google Scholar

[24]

J. Jahn, "Vector Optimization: Theory, Applications, and Extensions,", Springer-Verlag, (2004).   Google Scholar

[25]

J. Jahn, "Introduction to the Theory of Nonlinear Optimization,", 3rd edition, (2007).   Google Scholar

[26]

Y. Liu, K. L. Teo and R. P. Agarwal, A general approach to nonlinear multiple control problems with perturbation consideration,, Math. Comput. Modelling, 26 (1997), 49.  doi: 10.1016/S0895-7177(97)00239-2.  Google Scholar

[27]

D. T. Lųc, "Theory of Vector Optimization,", Lecture Notes in Economics and Mathematical Systems, 319 (1989).   Google Scholar

[28]

K. Miettinen, "Nonlinear Multiobjective Optimization,", International Series in Operations Research & Management Science, 12 (1999).  doi: 10.1007/978-1-4615-5563-6.  Google Scholar

[29]

J. Philip, Algorithms for the vector maximization problem,, Math. Programming, 2 (1972), 207.  doi: 10.1007/BF01584543.  Google Scholar

[30]

T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar

[31]

K. L. Teo, D. Li and Y. Liu, Perturbation feedback control in general multiple linear-quadratic control problems,, IMA J. Math. Control Inform., 15 (1998), 303.  doi: 10.1093/imamci/15.3.303.  Google Scholar

[32]

Y. Yamamoto, Optimization over the efficient set: Overview,, J. Global Optim., 22 (2002), 285.  doi: 10.1023/A:1013875600711.  Google Scholar

[1]

Hamed Fazlollahtabar, Mohammad Saidi-Mehrabad. Optimizing multi-objective decision making having qualitative evaluation. Journal of Industrial & Management Optimization, 2015, 11 (3) : 747-762. doi: 10.3934/jimo.2015.11.747
[2]

Adriel Cheng, Cheng-Chew Lim. Optimizing system-on-chip verifications with multi-objective genetic evolutionary algorithms. Journal of Industrial & Management Optimization, 2014, 10 (2) : 383-396. doi: 10.3934/jimo.2014.10.383
[3]

Yuan-mei Xia, Xin-min Yang, Ke-quan Zhao. A combined scalarization method for multi-objective optimization problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020088
[4]

Xia Zhao, Jianping Dou. Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1263-1288. doi: 10.3934/jimo.2018095
[5]

Liwei Zhang, Jihong Zhang, Yule Zhang. Second-order optimality conditions for cone constrained multi-objective optimization. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1041-1054. doi: 10.3934/jimo.2017089
[6]

Danthai Thongphiew, Vira Chankong, Fang-Fang Yin, Q. Jackie Wu. An on-line adaptive radiation therapy system for intensity modulated radiation therapy: An application of multi-objective optimization. Journal of Industrial & Management Optimization, 2008, 4 (3) : 453-475. doi: 10.3934/jimo.2008.4.453
[7]

Qiang Long, Xue Wu, Changzhi Wu. Non-dominated sorting methods for multi-objective optimization: Review and numerical comparison. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020009
[8]

Min Zhang, Gang Li. Multi-objective optimization algorithm based on improved particle swarm in cloud computing environment. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1413-1426. doi: 10.3934/dcdss.2019097
[9]

Han Yang, Jia Yue, Nan-jing Huang. Multi-objective robust cross-market mixed portfolio optimization under hierarchical risk integration. Journal of Industrial & Management Optimization, 2020, 16 (2) : 759-775. doi: 10.3934/jimo.2018177
[10]

Chaabane Djamal, Pirlot Marc. A method for optimizing over the integer efficient set. Journal of Industrial & Management Optimization, 2010, 6 (4) : 811-823. doi: 10.3934/jimo.2010.6.811
[11]

Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019130
[12]

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 & Management Optimization, 2017, 13 (3) : 1189-1211. doi: 10.3934/jimo.2016068
[13]

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 & Optimization, 2011, 1 (3) : 487-493. doi: 10.3934/naco.2011.1.487
[14]

Tien-Fu Liang, Hung-Wen Cheng. Multi-objective aggregate production planning decisions using two-phase fuzzy goal programming method. Journal of Industrial & Management Optimization, 2011, 7 (2) : 365-383. doi: 10.3934/jimo.2011.7.365
[15]

Zongmin Li, Jiuping Xu, Wenjing Shen, Benjamin Lev, Xiao Lei. Bilevel multi-objective construction site security planning with twofold random phenomenon. Journal of Industrial & Management Optimization, 2015, 11 (2) : 595-617. doi: 10.3934/jimo.2015.11.595
[16]

Lorena Rodríguez-Gallego, Antonella Barletta Carolina Cabrera, Carla Kruk, Mariana Nin, Antonio Mauttone. Establishing limits to agriculture and afforestation: A GIS based multi-objective approach to prevent algal blooms in a coastal lagoon. Journal of Dynamics & Games, 2019, 6 (2) : 159-178. doi: 10.3934/jdg.2019012
[17]

Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751
[18]

Jiahua Zhang, Shu-Cherng Fang, Yifan Xu, Ziteng Wang. A cooperative game with envy. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2049-2066. doi: 10.3934/jimo.2017031
[19]

Masoud Mohammadzadeh, Alireza Arshadi Khamseh, Mohammad Mohammadi. A multi-objective integrated model for closed-loop supply chain configuration and supplier selection considering uncertain demand and different performance levels. Journal of Industrial & Management Optimization, 2017, 13 (2) : 1041-1064. doi: 10.3934/jimo.2016061
[20]

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 & Management Optimization, 2019, 15 (2) : 855-879. doi: 10.3934/jimo.2018074

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences