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

Volumn 7, Issue 4, 2011, Pages 789-809

  Bonnel, Henri ^a   Pham, Ngoc Sang ^a  

^a UNIVERSITÉ DE LA NOUVELLE CALÉDONIE   (New Caledonia)

Author keywords

Grand coalition p player cooperative differential game; Multi objective Control and Optimization; Optimizing over the Pareto set

Indexed keywords

EID: 80053089708     PISSN: 15475816     EISSN: 1553166X     Source Type: Journal    
DOI: 10.3934/jimo.2011.7.789     Document Type: Article

References (32)

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.
2. L. T. H. An, P. D. Tao and L. D. Muu, Numerical solution for optimization over the ecient set by d.c. optimization algorithms, Oper. Res. Lett., 19 (1996), 117-128.
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 ecient set, J. Math. Anal. Appl., 98 (1984), 562-580.
6. H. P. Benson, A nite, nonadjacent extreme point search algorithm for optimization over the ecient set, J. Optim. Theory Appl., 73 (1992), 47-64.
7. S. Bolintineanu, Optimality conditions for minimization over the (weakly or properly) ecient set, J. Math. Anal. Appl., 173 (1993), 523-541.
8. S. Bolintineanu, Minimization of a quasi-concave function over an ecient set, Math. Programming, 61 (1993), 89-110.
9. S. Bolintineanu, Necessary conditions for nonlinear suboptimization over the weakly-ecient set, J. Optim. Theory Appl., 78 (1993), 579-598.
10. S. Bolintinéanu and M. El Maghri, Pénalisation dans l'optimisation sur l'ensemble faiblement ecient, (French) [Penalization in optimization over the weakly ecient set], RAIRO Rech. Opér., 31 (1997), 295-310.
11. H. Bonnel and C. Y. Kaya, Optimization over the ecient set of multi-objective convex optimal control problems, J. Optim. Theory Appl., 147 (2010), 93-112.
12. H. Bonnel and J. Morgan, Semivectorial bilevel optimization problem: Penalty approach, J. Optim. Theory Appl., 131 (2006), 365-382.
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 ecient set using an active constraint approach, Z. Oper. Res., 35 (1991), 185-195.
17. J. P. Dauer and T. A. Fosnaugh, Optimization over the ecient set, J. Global Optim., 7 (1995), 261-277.
18. G. Eichfelder, "Adaptive Scalarization Methods in Multiobjective Optimization," Springer-Verlag, Berlin, 2008.
19. J. Engwerda, Necessary and sucient conditions for Pareto optimal solution of cooperative dierential games, SIAM J. Control Optim., 48 (2010), 3859-3881.
20. J. Fülöp, A cutting plane algorithm for linear optimization over the ecient set, in "Generalized Convexity" (Pëcs, 1992), Lecture notes in Economics and Mathematical System, 405, Springer-Verlag, Berlin, (1994), 374-385.
21. A. Göpfert, H. Riahi, C. Tammer and C. Zǎlinescu, "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 ecient or weakly-ecient set, European J. Oper. Res., 117 (1999), 239-252.
23. R. Horst, N. V. Thoai, Y. Yamamoto and D. Zenke, On optimization over the ecient set in linear multicriteria programming, J. Optim. Theory Appl., 134 (2007), 433-443.
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.
27. D. T. Lu. c, "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.
29. J. Philip, Algorithms for the vector maximization problem, Math. Programming, 2 (1972), 207-229.
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.
32. Y. Yamamoto, Optimization over the ecient set: Overview, J. Global Optim., 22 (2002), 285-317.