Abstract
We consider the problem of optimal design of hybrid car engines which combine thermal and electric power. The optimal configuration of the different motors composing the hybrid system involves the choice of certain design parameters. For a given configuration, the goal is to minimize the fuel consumption along a trajectory. This is an optimal control problem with one state variable.
The simultaneous optimization of design parameters and trajectories can be formulated as a bilevel optimization problem. The lower level computes the optimal control for a given architecture. The higher level seeks for the optimal design parameters by solving a nonconvex nonsmooth optimization problem with a bundle method.
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M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equa-tions, Systems and Control: Foundations and Applications. Birkhäuser, Boston, 1997.
G. Barles, Solutions de viscositédes équations de Hamilton-Jacobi,vol. 17 of Mathématiques et applications. Springer, Paris, 1994.
G. Barles and P. E. Souganidis, “Convergence of approximation schemes for fully nonlinear second order equations,” Asymptotic Analysis vol. 4, pp. 271–283, 1991.
J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001.
G. Blanchon, J. C. Dodu, and J. F. Bonnans, “Optimisation des réseaux électriques de grande taille,” in Analysis and Optimization of Systems, A. Bensoussan and J. L. Lions (Eds.), vol. 144 of Lecture Notes in Information and Control Sciences, Springer-Verlag: Berlin, 1990.
J. F. Bonnans, J. Ch. Gilbert, C. Lemaréchal, and C. Sagastizábal, Numerical Optimization, Universitext. Springer-Verlag: Berlin, 2002.
J. F. Bonnans, Th. Guilbaud, and H. Zidani, “Bilevel optimization for parametric optimal control problems,” Working paper. F. H. Clarke, Optimization and Nonsmooth Analysis, J.Wiley: New York, 1983.
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag: Berlin, 1990.
J. Gauvin, “The generalized gradient of a marginal function in mathematical programming,” Mathematics of Operations Research vol. 4, pp. 458–463, 1979.
W. W. Hager, “Runge-Kutta methods in optimal control and the transformed adjoint system,” Numerische Mathematik vol. 87, pp. 247–282, 2000.
H. Ishii and S. Koike, “A new formulation of state constraint problems for first-order PDEs,” SIAM Journal on Control and Optimization vol. 34, no. 2, pp. 554–571, 1996.
H. J. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, 2nd edition, vol. 24 of Applications of Mathematics. Springer: New York. 2001.
C. Lemaréchal and C. Sagastizábal, “Variable metric bundle methods: From conceptual to implementable forms,” Mathematical Programming vol. 76, pp. 393–410, 1997.
C. Lemaréchal, J.-J. Strodiot, and A. Bihain, “On a bundle method for nonsmooth optimization,” in Nonlinear Programming,O. Mangasarian, R. R. Meyer, and S. M. Robinson (Eds.), vol. 4, Academic Press, 1981, pp. 245–282.
P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations,vol. 69 of Research Notes in Mathematics. Pitman, Boston, 1982.
L. Lukšan and J. Vlček, “A bundle-Newton method for nonsmooth unconstrained minimization,” Math. Programming vol. 83, no. 3, Ser. A, pp. 373–391, 1998.
D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica J. IFAC,vol. 36, no. 6, pp. 789–814, 2000. See also the correction in Automatica vol. 37, no. 3, p. 483, 2001.
R. Mifflin, “Semismooth and semiconvex functions in constrained optimization,” SIAM J. Control and Optimization vol. 15, pp. 959–972, 1977.
R. Mifflin, “A modification and extension of Lemarechal's algorithm for nonsmooth minimization,” Math. Programming Stud. vol. 17, pp. 77–90, 1982.
J. Outrata, M. Kočvara, and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers: Boston, 1998.
H. Schramm and J. Zowe, “A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results,” SIAM J. on Optimization vol. 2, no. 1, pp. 121–152, 1992.
H. Sonnet, Dictionnaire des Mathématiques Appliquées, Librairie de L. Hachette et Cie, Paris, 1867.
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Bonnans, J.F., Guilbaud, T., Ketfi-Cherif, A. et al. Parametric Optimization of Hybrid Car Engines. Optimization and Engineering 5, 395–415 (2004). https://doi.org/10.1023/B:OPTE.0000042032.47856.e5
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DOI: https://doi.org/10.1023/B:OPTE.0000042032.47856.e5