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Bregman-Golden Ratio Algorithms for Variational Inequalities

Author

Listed:
  • Matthew K. Tam

    (The University of Melbourne)

  • Daniel J. Uteda

    (The University of Melbourne)

Abstract

Variational inequalities provide a framework through which many optimisation problems can be solved, in particular, saddle-point problems. In this paper, we study modifications to the so-called Golden RAtio ALgorithm (GRAAL) for variational inequalities—a method which uses a fully explicit adaptive step-size and provides convergence results under local Lipschitz assumptions without requiring backtracking. We present and analyse two Bregman modifications to GRAAL: the first uses a fixed step size and converges under global Lipschitz assumptions, and the second uses an adaptive step-size rule. Numerical performance of the former method is demonstrated on a bimatrix game arising in network communication, and of the latter on two problems, namely, power allocation in Gaussian communication channels and N-person Cournot completion games. In all of these applications, an appropriately chosen Bregman distance simplifies the projection steps computed as part of the algorithm.

Suggested Citation

  • Matthew K. Tam & Daniel J. Uteda, 2023. "Bregman-Golden Ratio Algorithms for Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 199(3), pages 993-1021, December.
    • Handle: RePEc:spr:joptap:v:199:y:2023:i:3:d:10.1007_s10957-023-02320-2
    DOI: 10.1007/s10957-023-02320-2
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