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

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  • 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.
    Mathematics Subject Classification: Primary: 90C29, 91A23, 49K30; Secondary: 49N10, 91A12.


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  • [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.


    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.


    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.


    V. Azhmyakov, An approach to controlled mechanical systems based on the multiobjective optimization technique, J. Ind. Manag. Optim., 4 (2008), 697-712.


    H. P. Benson, Optimization over the efficient set, J. Math. Anal. Appl., 98 (1984), 562-580.doi: 10.1016/0022-247X(84)90269-5.


    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.


    S. Bolintineanu, Optimality conditions for minimization over the (weakly or properly) efficient set, J. Math. Anal. Appl., 173 (1993), 523-541.


    S. Bolintineanu, Minimization of a quasi-concave function over an efficient set, Math. Programming, 61 (1993), 89-110.doi: 10.1007/BF01582141.


    S. Bolintineanu, Necessary conditions for nonlinear suboptimization over the weakly-efficient set, J. Optim. Theory Appl., 78 (1993), 579-598.doi: 10.1007/BF00939883.


    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.


    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.


    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.


    H. Bonnel, Optimality conditions for the semivectorial bilevel optimization problem, Pac. J. Optim., 2 (2006), 447-467.


    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.


    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.


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


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


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


    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.


    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.


    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.


    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.


    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.


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


    J. Jahn, "Introduction to the Theory of Nonlinear Optimization," 3rd edition, Springer, Berlin, 2007.


    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.


    D. T. Lųc, "Theory of Vector Optimization," Lecture Notes in Economics and Mathematical Systems, 319, Springer-Verlag, Berlin, 1989.


    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.


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


    T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, New Jersey, 1970.


    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.


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

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