# American Institute of Mathematical Sciences

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

 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.
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 and 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. [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. [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. [4] V. Azhmyakov, An approach to controlled mechanical systems based on the multiobjective optimization technique, J. Ind. Manag. Optim., 4 (2008), 697-712. [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. [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. [7] S. Bolintineanu, Optimality conditions for minimization over the (weakly or properly) efficient set, J. Math. Anal. Appl., 173 (1993), 523-541. [8] S. Bolintineanu, Minimization of a quasi-concave function over an efficient set, Math. Programming, 61 (1993), 89-110. doi: 10.1007/BF01582141. [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. [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. [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. [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. [13] H. Bonnel, Optimality conditions for the semivectorial bilevel optimization problem, Pac. J. Optim., 2 (2006), 447-467. [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. [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. [16] J. P. Dauer, Optimization over the efficient set using an active constraint approach, Z. Oper. Res., 35 (1991), 185-195. [17] J. P. Dauer and T. A. Fosnaugh, Optimization over the efficient set, J. Global Optim., 7 (1995), 261-277. doi: 10.1007/BF01279451. [18] G. Eichfelder, "Adaptive Scalarization Methods in Multiobjective Optimization," Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-79159-1. [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. [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. [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. [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. [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. [24] J. Jahn, "Vector Optimization: Theory, Applications, and Extensions," Springer-Verlag, Berlin, 2004. [25] J. Jahn, "Introduction to the Theory of Nonlinear Optimization," 3rd edition, Springer, Berlin, 2007. [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. [27] D. T. Lųc, "Theory of Vector Optimization," Lecture Notes in Economics and Mathematical Systems, 319, Springer-Verlag, Berlin, 1989. [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. [29] J. Philip, Algorithms for the vector maximization problem, Math. Programming, 2 (1972), 207-229. doi: 10.1007/BF01584543. [30] T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, New Jersey, 1970. [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. [32] Y. Yamamoto, Optimization over the efficient set: Overview, J. Global Optim., 22 (2002), 285-317. doi: 10.1023/A:1013875600711.

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. [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. [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. [4] V. Azhmyakov, An approach to controlled mechanical systems based on the multiobjective optimization technique, J. Ind. Manag. Optim., 4 (2008), 697-712. [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. [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. [7] S. Bolintineanu, Optimality conditions for minimization over the (weakly or properly) efficient set, J. Math. Anal. Appl., 173 (1993), 523-541. [8] S. Bolintineanu, Minimization of a quasi-concave function over an efficient set, Math. Programming, 61 (1993), 89-110. doi: 10.1007/BF01582141. [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. [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. [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. [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. [13] H. Bonnel, Optimality conditions for the semivectorial bilevel optimization problem, Pac. J. Optim., 2 (2006), 447-467. [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. [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. [16] J. P. Dauer, Optimization over the efficient set using an active constraint approach, Z. Oper. Res., 35 (1991), 185-195. [17] J. P. Dauer and T. A. Fosnaugh, Optimization over the efficient set, J. Global Optim., 7 (1995), 261-277. doi: 10.1007/BF01279451. [18] G. Eichfelder, "Adaptive Scalarization Methods in Multiobjective Optimization," Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-79159-1. [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. [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. [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. [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. [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. [24] J. Jahn, "Vector Optimization: Theory, Applications, and Extensions," Springer-Verlag, Berlin, 2004. [25] J. Jahn, "Introduction to the Theory of Nonlinear Optimization," 3rd edition, Springer, Berlin, 2007. [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. [27] D. T. Lųc, "Theory of Vector Optimization," Lecture Notes in Economics and Mathematical Systems, 319, Springer-Verlag, Berlin, 1989. [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. [29] J. Philip, Algorithms for the vector maximization problem, Math. Programming, 2 (1972), 207-229. doi: 10.1007/BF01584543. [30] T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, New Jersey, 1970. [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. [32] Y. Yamamoto, Optimization over the efficient set: Overview, J. Global Optim., 22 (2002), 285-317. doi: 10.1023/A:1013875600711.
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