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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) |
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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