# Golden Ratio Algorithms for Variational Inequalities

arXiv: https://arxiv.org/abs/1803.08832 · HTML: https://mlsanity.com/paper/1803.08832/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Mar 2018.

## Built on by (most cited first, 30 of 38)

- [Faster Single-loop Algorithms for Minimax Optimization without Strong Concavity](https://mlsanity.com/paper/2112.05604/) (2021)
- [The Complexity of Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2103.15888/) (2021)
- [Accelerated Single-Call Methods for Constrained Min-Max Optimization](https://mlsanity.com/paper/2210.03096/) (2022)
- [Halpern-Type Accelerated and Splitting Algorithms For Monotone Inclusions](https://mlsanity.com/paper/2110.08150/) (2021)
- [Adaptive Proximal Gradient Method for Convex Optimization](https://mlsanity.com/paper/2308.02261/) (2023)
- [No-Regret Dynamics in the Fenchel Game: A Unified Framework for Algorithmic Convex Optimization](https://mlsanity.com/paper/2111.11309/) (2021)
- [Accelerated Algorithms for Constrained Nonconvex-Nonconcave Min-Max Optimization and Comonotone Inclusion](https://mlsanity.com/paper/2206.05248/) (2022)
- [Adaptive proximal algorithms for convex optimization under local Lipschitz continuity of the gradient](https://mlsanity.com/paper/2301.04431/) (2023)
- [A simple uniformly optimal method without line search for convex optimization](https://mlsanity.com/paper/2310.10082/) (2023)
- [Nest Your Adaptive Algorithm for Parameter-Agnostic Nonconvex Minimax Optimization](https://mlsanity.com/paper/2206.00743/) (2022)
- [Beyond the Golden Ratio for Variational Inequality Algorithms](https://mlsanity.com/paper/2212.13955/) (2022)
- [Variance Reduced Halpern Iteration for Finite-Sum Monotone Inclusions](https://mlsanity.com/paper/2310.02987/) (2023)
- [Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm](https://mlsanity.com/paper/2202.06950/) (2022)
- [Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization](https://mlsanity.com/paper/2303.03900/) (2023)
- [Adaptive and Optimal Second-order Optimistic Methods for Minimax Optimization](https://mlsanity.com/paper/2406.02016/) (2024)
- [A first-order primal-dual method with adaptivity to local smoothness](https://mlsanity.com/paper/2110.15148/) (2021)
- [Sublinear Convergence Rates of Extragradient-Type Methods: A Survey on Classical and Recent Developments](https://mlsanity.com/paper/2303.17192/) (2023)
- [Nesterov Finds GRAAL: Optimal and Adaptive Gradient Method for Convex Optimization](https://mlsanity.com/paper/2507.09823/) (2025)
- [Cyclic Coordinate Dual Averaging with Extrapolation](https://mlsanity.com/paper/2102.13244/) (2021)
- [Primal Methods for Variational Inequality Problems with Functional Constraints](https://mlsanity.com/paper/2403.12859/) (2024)
- [Unbiased and Biased Variance-Reduced Forward-Reflected-Backward Splitting Methods for Stochastic Composite Inclusions](https://mlsanity.com/paper/2603.15576/) (2026)
- [Distributed Extra-gradient with Optimal Complexity and Communication Guarantees](https://mlsanity.com/paper/2308.09187/) (2023)
- [On the convergence of adaptive first order methods: proximal gradient and alternating minimization algorithms](https://mlsanity.com/paper/2311.18431/) (2023)
- [Auto-conditioned primal-dual hybrid gradient method and alternating direction method of multipliers](https://mlsanity.com/paper/2410.01979/) (2024)
- [Semi-Implicit Hybrid Gradient Methods with Application to Adversarial Robustness](https://mlsanity.com/paper/2202.10523/) (2022)
- [Primal-dual extrapolation methods for monotone inclusions under local Lipschitz continuity](https://mlsanity.com/paper/2206.00973/) (2022)
- [Layer-wise Quantization for Quantized Optimistic Dual Averaging](https://mlsanity.com/paper/2505.14371/) (2025)
- [Stochastic variance reduced extragradient methods for solving hierarchical variational inequalities](https://mlsanity.com/paper/2602.13510/) (2026)
- [Parameter-Free Non-Ergodic Extragradient Algorithms for Solving Monotone Variational Inequalities](https://mlsanity.com/paper/2604.07662/) (2026)
- [Extragradient-Type Methods with O (1/k) Last-Iterate Convergence Rates for Co-Hypomonotone Inclusions](https://mlsanity.com/paper/2302.04099/) (2023)

## Abstract (arXiv)

The paper presents a fully explicit algorithm for monotone variational inequalities. The method uses variable stepsizes that are computed using two previous iterates as an approximation of the local Lipschitz constant without running a linesearch. Thus, each iteration of the method requires only one evaluation of a monotone operator F and a proximal mapping g. The operator F need not be Lipschitz-continuous, which also makes the algorithm interesting in the area of composite minimization where one cannot use the descent lemma. The method exhibits an ergodic O(1/k) convergence rate and R-linear rate, if F, g satisfy the error bound condition. We discuss possible applications of the method to fixed point problems. We discuss possible applications of the method to fixed point problems as well as its different generalizations.