American Institute of Mathematical Sciences

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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, Birkhäuser Verlag, Basel, 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), 117-128. 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), A Wiley-Interscience Publication, John Wiley & Sons, Inc., 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), 697-712.  Google Scholar

[5]

H. P. Benson, Optimization over the efficient set, J. Math. Anal. Appl., 98 (1984), 562-580. 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), 47-64. 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), 523-541.  Google Scholar

[8]

S. Bolintineanu, Minimization of a quasi-concave function over an efficient set, Math. Programming, 61 (1993), 89-110. 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), 579-598. 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], RAIRO Rech. Opér., 31 (1997), 295-310.  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), 93-112. 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), 365-382. 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), 447-467.  Google Scholar

[14]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

[15]

B. D. Craven, Aspects of multicriteria optimization, in "Recent Developments in Mathematical Programming" (ed. S. Kumar), Gordon and Breach Science Publishers, Philadelphia, (1991), 93-100. Google Scholar

[16]

J. P. Dauer, Optimization over the efficient set using an active constraint approach, Z. Oper. Res., 35 (1991), 185-195.  Google Scholar

[17]

J. P. Dauer and T. A. Fosnaugh, Optimization over the efficient set, J. Global Optim., 7 (1995), 261-277. doi: 10.1007/BF01279451.  Google Scholar

[18]

G. Eichfelder, "Adaptive Scalarization Methods in Multiobjective Optimization," Springer-Verlag, Berlin, 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), 3859-3881. doi: 10.1137/080726227.  Google Scholar

[20]

J. Fülöp, A cutting plane algorithm for linear optimization over the efficient set, in "Generalized Convexity" (Pécs, 1992), Lecture notes in Economics and Mathematical System, 405, Springer-Verlag, Berlin, (1994), 374-385.  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, Springer-Verlag, New York, 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), 239-252. 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), 433-443. doi: 10.1007/s10957-007-9219-8.  Google Scholar

[24]

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

[25]

J. Jahn, "Introduction to the Theory of Nonlinear Optimization," 3rd edition, Springer, Berlin, 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), 49-58. 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, Springer-Verlag, Berlin, 1989.  Google Scholar

[28]

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

[29]

J. Philip, Algorithms for the vector maximization problem, Math. Programming, 2 (1972), 207-229. doi: 10.1007/BF01584543.  Google Scholar

[30]

T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, New Jersey, 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), 303-315. doi: 10.1093/imamci/15.3.303.  Google Scholar

[32]

Y. Yamamoto, Optimization over the efficient set: Overview, J. Global Optim., 22 (2002), 285-317. 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, Birkhäuser Verlag, Basel, 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), 117-128. 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), A Wiley-Interscience Publication, John Wiley & Sons, Inc., 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), 697-712.  Google Scholar

[5]

H. P. Benson, Optimization over the efficient set, J. Math. Anal. Appl., 98 (1984), 562-580. 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), 47-64. 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), 523-541.  Google Scholar

[8]

S. Bolintineanu, Minimization of a quasi-concave function over an efficient set, Math. Programming, 61 (1993), 89-110. 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), 579-598. 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], RAIRO Rech. Opér., 31 (1997), 295-310.  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), 93-112. 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), 365-382. 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), 447-467.  Google Scholar

[14]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

[15]

B. D. Craven, Aspects of multicriteria optimization, in "Recent Developments in Mathematical Programming" (ed. S. Kumar), Gordon and Breach Science Publishers, Philadelphia, (1991), 93-100. Google Scholar

[16]

J. P. Dauer, Optimization over the efficient set using an active constraint approach, Z. Oper. Res., 35 (1991), 185-195.  Google Scholar

[17]

J. P. Dauer and T. A. Fosnaugh, Optimization over the efficient set, J. Global Optim., 7 (1995), 261-277. doi: 10.1007/BF01279451.  Google Scholar

[18]

G. Eichfelder, "Adaptive Scalarization Methods in Multiobjective Optimization," Springer-Verlag, Berlin, 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), 3859-3881. doi: 10.1137/080726227.  Google Scholar

[20]

J. Fülöp, A cutting plane algorithm for linear optimization over the efficient set, in "Generalized Convexity" (Pécs, 1992), Lecture notes in Economics and Mathematical System, 405, Springer-Verlag, Berlin, (1994), 374-385.  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, Springer-Verlag, New York, 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), 239-252. 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), 433-443. doi: 10.1007/s10957-007-9219-8.  Google Scholar

[24]

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

[25]

J. Jahn, "Introduction to the Theory of Nonlinear Optimization," 3rd edition, Springer, Berlin, 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), 49-58. 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, Springer-Verlag, Berlin, 1989.  Google Scholar

[28]

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

[29]

J. Philip, Algorithms for the vector maximization problem, Math. Programming, 2 (1972), 207-229. doi: 10.1007/BF01584543.  Google Scholar

[30]

T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, New Jersey, 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), 303-315. doi: 10.1093/imamci/15.3.303.  Google Scholar

[32]

Y. Yamamoto, Optimization over the efficient set: Overview, J. Global Optim., 22 (2002), 285-317. doi: 10.1023/A:1013875600711.  Google Scholar

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