Bregman-Golden Ratio Algorithms for Variational Inequalities

Matthew K. Tam () and Daniel J. Uteda ()
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Matthew K. Tam: The University of Melbourne
Daniel J. Uteda: The University of Melbourne

Journal of Optimization Theory and Applications, 2023, vol. 199, issue 3, No 6, 993-1021

Abstract: 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.

Keywords: Variational inequality; Saddle-point problems; Bregman distance; Local Lipschitz; Adaptive step size; 47J20; 49J40; 65K15; 65Y20 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10957-023-02320-2

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