Convergence Results for Proximal-Type Algorithm Beyond the Classical Golden-Ratio Constant for Equilibrium Problems

The golden-ratio algorithm has recently emerged as an elegant approach for solving variational inequality problems, employing a parameter (Formula presented.) traditionally restricted to the interval (Formula presented.). In this work, we prove that the admissible range of (Formula presented.) can be extended to (Formula presented.) within the broader framework of equilibrium problems, a genuine generalization that reveals the classical golden-ratio bound to be a special case rather than an intrinsic limitation. Alongside this parameter-space extension, we introduce a novel self-adaptive step-size strategy that generates step sizes becoming nondecreasing after finitely many iterations, fundamentally departing from the nonincreasing step sizes required by all existing methods. This new step-size dynamics enables interesting convergence guarantees: We establish weak convergence under pseudo-monotonicity, and under strong pseudo-monotonicity, we prove linear convergence for (Formula presented.) in the classical range and strong convergence for (Formula presented.) in the extended range, results unavailable in previous golden-ratio algorithms. Each iteration requires only a single proximal evaluation, maintaining computational efficiency. The efficacy of the proposed method is demonstrated through comprehensive numerical experiments, including an application to dynamic user optimal equilibrium problems in traffic networks, with explicit examples showing superior performance on ill-conditioned and infinite-dimensional problems where classical methods struggle.