Bregman-Golden Ratio Algorithms for Variational Inequalities

• Published in: Journal of optimization theory and applications Vol. 199; no. 3; pp. 993 - 1021
• Main Authors: Tam, Matthew K., Uteda, Daniel J.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2023; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Communication; Convergence; Engineering; Inequalities; Lipschitz condition; Mathematical functions; Mathematics; Mathematics and Statistics; Methods; Operations Research/Decision Theory; Optimization; Saddle points; Theory of Computation; Variational methods; 65K15; 47J20; Variational inequality; 49J40; Adaptive step size; 65Y20; Saddle-point problems; Bregman distance; Local Lipschitz
• ISSN: 0022-3239; 1573-2878; 1573-2878
• DOI: 10.1007/s10957-023-02320-2

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.