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Jointly Convex Generalized Nash Equilibria and Elliptic Multiobjective Optimal Control

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Abstract

We deal with jointly convex generalized Nash equilibrium problems in infinite-dimensional spaces. For their solution, we extend a finite-dimensional optimization approach and design a convergent algorithm in Hilbert space. Then we apply our investigations to a class of multiobjective optimal control problems with control and state constraints that are governed by elliptic partial differential equations. We present a new reformulation as a jointly convex generalized Nash equilibrium problem. We study a finite element approximation of such a multiobjective optimal control problem, and further we prove convergence in appropriate function spaces. Finally, we provide some numerical results that show the effectiveness of our algorithm for multiobjective optimal control problems.

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References

  1. Ramos, A.M., Glowinski, R., Periaux, J.: Nash equilibria for the multiobjective control of linear partial differential equations. J. Optim. Theory Appl. 112, 457–498 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Tang, Z., Desideri, J.-A., Periaux, J.: Multicriterion aerodynamic shape design optimization and inverse problems using control theory and Nash games. J. Optim. Theory Appl. 135, 599–622 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hintermüller, M., Surowiec, T., Kämmler, A.: Generalized Nash Equilibrium Problems in Banach Spaces: Theory, Nikaido–Isoda-Based Path-Following Methods, and Applications. Preprint, Humbold University Berlin (2014)

  4. Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  5. Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. 4OR 5, 173–210 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Tröltzsch, F.: Optimal Control of Partial Differential Equations. Grad. Stud. Math. 112, AMS, Providence, RI, (2010). Translated from the 2005 German original by J. Sprekels

  7. Roubíček, T.: Noncooperative games with elliptic systems. Optimal control of partial differential equations (Chemnitz, 1998). Int. Ser. Numer. Math. 133, 245–255 (1999)

    Google Scholar 

  8. Ramos, A.M.: Numerical methods for Nash equilibria in multiobjective control of partial differential equations. In: Barbu, V., et al. (eds.) Analysis and Optimization of Differential Systems, pp. 333–344. Springer, New York (2003)

    Chapter  Google Scholar 

  9. Borzi, A., Kanzow, C.: Formulation and numerical solution of Nash equilibrium multiobjective elliptic control problems. SIAM J. Control Optim. 51, 718–744 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hintermüller, M., Surowiec, T.: A PDE-constrained generalized Nash equilibrium problem with pointwise control and state constraints. Pac. J. Optim. 9(2), 251–273 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Dreves, A.: A Nash equilibrium approach for multiobjective optimal control problems with elliptic partial differential equations. Technical Report, Universität der Bundeswehr München (2014)

  12. Rosen, J.B.: Existence and uniqueness of equilibrium points for concave \(N\)-person games. Econometrica 33, 520–534 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  13. Pang, J.-S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2, 21–56 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dreves, A., von Heusinger, A., Kanzow, C., Fukushima, M.: A globalized Newton method for the computation of normalized Nash equilibria. J. Global Optim. 56, 327–340 (2010)

    Article  Google Scholar 

  15. von Heusinger, A., Kanzow, C.: Relaxation methods for generalized Nash equilibrium problems with inexact line search. J. Optim. Theory Appl. 109, 159–183 (2009)

    Article  Google Scholar 

  16. Uryasev, S., Rubinstein, R.Y.: On relaxation algorithms in computation of noncooperative equilibria. IEEE Trans. Autom. Control 39, 1263–1267 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nikaido, H., Isoda, K.: Note on noncooperative convex games. Pac. J. Math. 5, 807–815 (1955)

    Article  MathSciNet  Google Scholar 

  18. Flåm, S.D., Ruszczyński, R.: Noncooperative convex games: computing equilibrium by partial regularization. IIASA working paper 94–42, Laxenburg, Austria (1994)

  19. Gürkan, G., Pang, J.-S.: Approximations of Nash equilibria. Math. Program. 117, 223–253 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. von Heusinger, A., Kanzow, C.: Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions. Comput. Optim. Appl. 43, 353–377 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer Series in Operations Research, New York (2000)

    Book  MATH  Google Scholar 

  22. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. 2 edn., Pure Appl. Math., vol. 140, Academic Press, New York (2003)

  23. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, New York (1983)

    Book  MATH  Google Scholar 

  24. Meyer, C.: Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybern. 37, 51–83 (2008)

    MATH  Google Scholar 

  25. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)

    Book  MATH  Google Scholar 

  26. Gwinner, J.: A class of random variational inequalities and simple random unilateral boundary value problems: Existence, discretization, finite element approximation. Stoch. Anal. Appl. 18, 967–993 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications. Academic Press, New York (1980)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Aerospace Engineering, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, Neubiberg, 85577, Germany

    Axel Dreves & Joachim Gwinner

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  1. Axel Dreves
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  2. Joachim Gwinner
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Correspondence to Axel Dreves.

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The authors declare that they have no conflict of interest.

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Communicated by Negash G. Medhin.

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Dreves, A., Gwinner, J. Jointly Convex Generalized Nash Equilibria and Elliptic Multiobjective Optimal Control. J Optim Theory Appl 168, 1065–1086 (2016). https://doi.org/10.1007/s10957-015-0788-7

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  • DOI: https://doi.org/10.1007/s10957-015-0788-7

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