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Jointly Convex Generalized Nash Equilibria and Elliptic Multiobjective Optimal Control

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  • Axel Dreves

    (Universität der Bundeswehr München)

  • Joachim Gwinner

    (Universität der Bundeswehr München)

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.

Suggested Citation

  • Axel Dreves & Joachim Gwinner, 2016. "Jointly Convex Generalized Nash Equilibria and Elliptic Multiobjective Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 1065-1086, March.
    • Handle: RePEc:spr:joptap:v:168:y:2016:i:3:d:10.1007_s10957-015-0788-7
    DOI: 10.1007/s10957-015-0788-7
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    References listed on IDEAS

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    1. A. Heusinger & C. Kanzow, 2009. "Relaxation Methods for Generalized Nash Equilibrium Problems with Inexact Line Search," Journal of Optimization Theory and Applications, Springer, vol. 143(1), pages 159-183, October.
    2. A.M. Ramos & R. Glowinski & J. Periaux, 2002. "Nash Equilibria for the Multiobjective Control of Linear Partial Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 112(3), pages 457-498, March.
    3. Anna Heusinger & Christian Kanzow, 2009. "Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions," Computational Optimization and Applications, Springer, vol. 43(3), pages 353-377, July.
    4. Z. Tang & J.-A. Désidéri & J. Périaux, 2007. "Multicriterion Aerodynamic Shape Design Optimization and Inverse Problems Using Control Theory and Nash Games," Journal of Optimization Theory and Applications, Springer, vol. 135(3), pages 599-622, December.
    5. Jong-Shi Pang & Masao Fukushima, 2005. "Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games," Computational Management Science, Springer, vol. 2(1), pages 21-56, January.
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