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An explicit Tikhonov algorithm for nested variational inequalities

Author

Listed:
  • Lorenzo Lampariello

    (Roma Tre University)

  • Christoph Neumann

    (Karlsruhe Institute of Technology (KIT))

  • Jacopo M. Ricci

    (Roma Tre University)

  • Simone Sagratella

    (Sapienza University of Rome)

  • Oliver Stein

    (Karlsruhe Institute of Technology (KIT))

Abstract

We consider nested variational inequalities consisting in a (upper-level) variational inequality whose feasible set is given by the solution set of another (lower-level) variational inequality. Purely hierarchical convex bilevel optimization problems and certain multi-follower games are particular instances of nested variational inequalities. We present an explicit and ready-to-implement Tikhonov-type solution method for such problems. We give conditions that guarantee the convergence of the proposed method. Moreover, inspired by recent works in the literature, we provide a convergence rate analysis. In particular, for the simple bilevel instance, we are able to obtain enhanced convergence results.

Suggested Citation

  • Lorenzo Lampariello & Christoph Neumann & Jacopo M. Ricci & Simone Sagratella & Oliver Stein, 2020. "An explicit Tikhonov algorithm for nested variational inequalities," Computational Optimization and Applications, Springer, vol. 77(2), pages 335-350, November.
  • Handle: RePEc:spr:coopap:v:77:y:2020:i:2:d:10.1007_s10589-020-00210-1
    DOI: 10.1007/s10589-020-00210-1
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    References listed on IDEAS

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    1. Francisco Facchinei & Lorenzo Lampariello, 2011. "Partial penalization for the solution of generalized Nash equilibrium problems," Journal of Global Optimization, Springer, vol. 50(1), pages 39-57, May.
    2. Giuseppe Marino & Hong-Kun Xu, 2011. "Explicit Hierarchical Fixed Point Approach to Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 149(1), pages 61-78, April.
    3. Simone Sagratella, 2017. "Computing equilibria of Cournot oligopoly models with mixed-integer quantities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(3), pages 549-565, December.
    4. Francisco Facchinei & Christian Kanzow & Sebastian Karl & Simone Sagratella, 2015. "The semismooth Newton method for the solution of quasi-variational inequalities," Computational Optimization and Applications, Springer, vol. 62(1), pages 85-109, September.
    5. Didier Aussel & Simone Sagratella, 2017. "Sufficient conditions to compute any solution of a quasivariational inequality via a variational inequality," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(1), pages 3-18, February.
    6. Lorenzo Lampariello & Simone Sagratella, 2020. "Numerically tractable optimistic bilevel problems," Computational Optimization and Applications, Springer, vol. 76(2), pages 277-303, June.
    7. Simone Sagratella, 2017. "Algorithms for generalized potential games with mixed-integer variables," Computational Optimization and Applications, Springer, vol. 68(3), pages 689-717, December.
    8. Lorenzo Lampariello & Simone Sagratella, 2017. "A Bridge Between Bilevel Programs and Nash Games," Journal of Optimization Theory and Applications, Springer, vol. 174(2), pages 613-635, August.
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    Cited by:

    1. Giancarlo Bigi & Lorenzo Lampariello & Simone Sagratella & Valerio Giuseppe Sasso, 2023. "Approximate variational inequalities and equilibria," Computational Management Science, Springer, vol. 20(1), pages 1-16, December.
    2. Lampariello, Lorenzo & Neumann, Christoph & Ricci, Jacopo M. & Sagratella, Simone & Stein, Oliver, 2021. "Equilibrium selection for multi-portfolio optimization," European Journal of Operational Research, Elsevier, vol. 295(1), pages 363-373.
    3. Lorenzo Lampariello & Gianluca Priori & Simone Sagratella, 2022. "On the solution of monotone nested variational inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(3), pages 421-446, December.

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