# Projected Reflected Gradient Methods for Monotone Variational Inequalities

arXiv: https://arxiv.org/abs/1502.04968 · HTML: https://mlsanity.com/paper/1502.04968/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Feb 2015.

## Built on by (most cited first, 27 of 27)

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- [Tight Last-Iterate Convergence of the Extragradient and the Optimistic Gradient Descent-Ascent Algorithm for Constrained Monotone Variational Inequalities](https://mlsanity.com/paper/2204.09228/) (2022)
- [Generalized Optimistic Methods for Convex-Concave Saddle Point Problems](https://mlsanity.com/paper/2202.09674/) (2022)
- [The Last-Iterate Convergence Rate of Optimistic Mirror Descent in Stochastic Variational Inequalities](https://mlsanity.com/paper/2107.01906/) (2021)
- [Beyond the Golden Ratio for Variational Inequality Algorithms](https://mlsanity.com/paper/2212.13955/) (2022)
- [Convergence Rates of Two-Time-Scale Gradient Descent-Ascent Dynamics for Solving Nonconvex Min-Max Problems](https://mlsanity.com/paper/2112.09579/) (2021)
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- [Randomized Block-Coordinate Optimistic Gradient Algorithms for Root-Finding Problems](https://mlsanity.com/paper/2301.03113/) (2023)
- [DIPPA: An improved Method for Bilinear Saddle Point Problems](https://mlsanity.com/paper/2103.08270/) (2021)
- [Sublinear Convergence Rates of Extragradient-Type Methods: A Survey on Classical and Recent Developments](https://mlsanity.com/paper/2303.17192/) (2023)
- [Expected Variational Inequalities](https://mlsanity.com/paper/2502.18605/) (2025)
- [Solving Constrained Variational Inequalities via a First-order Interior Point-based Method](https://mlsanity.com/paper/2206.10575/) (2022)
- [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)
- [An Inexact Halpern Iteration with Application to Distributionally Robust Optimization](https://mlsanity.com/paper/2402.06033/) (2024)
- [Multiscale replay: A robust algorithm for stochastic variational inequalities with a Markovian buffer](https://mlsanity.com/paper/2601.01502/) (2026)
- [Extragradient Type Methods for Riemannian Variational Inequality Problems](https://mlsanity.com/paper/2309.14155/) (2023)
- [Layer-wise Quantization for Quantized Optimistic Dual Averaging](https://mlsanity.com/paper/2505.14371/) (2025)
- [A Variational Inequality Approach to Bayesian Regression Games](https://mlsanity.com/paper/2103.13509/) (2021)
- [Extragradient-Type Methods with O (1/k) Last-Iterate Convergence Rates for Co-Hypomonotone Inclusions](https://mlsanity.com/paper/2302.04099/) (2023)
- [Near-Optimal Distributed Minimax Optimization under the Second-Order Similarity](https://mlsanity.com/paper/2405.16126/) (2024)
- [Shuffling Heuristic in Variational Inequalities: Establishing New Convergence Guarantees](https://mlsanity.com/paper/2509.04133/) (2025)
- [Last-Iterate Convergence of Optimistic Multiplicative Weight Update](https://mlsanity.com/paper/2606.11773/) (2026)
- [Second-Moment Stochastic Approximation Methods](https://mlsanity.com/paper/2609.36600/) (2026)

## Abstract (arXiv)

This paper is concerned with some new projection methods for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. First, we propose the projected reflected gradient algorithm with a constant stepsize. It is similar to the projected gradient method, namely, the method requires only one projection onto the feasible set and only one value of the mapping per iteration. This distinguishes our method from most other projection-type methods for variational inequalities with monotone mapping. Also we prove that it has R-linear rate of convergence under the strong monotonicity assumption. The usual drawback of algorithms with constant stepsize is the requirement to know the Lipschitz constant of the mapping. To avoid this, we modify our first algorithm so that the algorithm needs at most two projections per iteration. In fact, our computational experience shows that such cases with two projections are very rare. This scheme, at least theoretically, seems to be very effective. All methods are shown to be globally convergent to a solution of the variational inequality. Preliminary results from numerical experiments are quite promising.