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In this paper, we introduce two golden ratio algorithms with new stepsize rules for solving pseudomonotone and Lipschitz variational inequalities in finite dimensional Hilbert spaces. The presented stepsize rules allow the resulting…

Optimization and Control · Mathematics 2019-04-17 Dang Van Hieu , Yeol Je Cho , Yi-bin Xiao

We improve the understanding of the $\textit{golden ratio algorithm}$, which solves monotone variational inequalities (VI) and convex-concave min-max problems via the distinctive feature of adapting the step sizes to the local Lipschitz…

Optimization and Control · Mathematics 2022-12-29 Ahmet Alacaoglu , Axel Böhm , Yura Malitsky

In this paper an explicit algorithm is proposed for solving an equilibrium problem whose associated bifunction is pseudomonotone and satisfies a Lipschitz-type condition. Contrary to many algorithms, our algorithm is done without using…

Optimization and Control · Mathematics 2019-07-10 Dang Van Hieu , Jean Jacques Strodiot , Le Dung Muu

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…

Optimization and Control · Mathematics 2022-08-11 Matthew K. Tam , Daniel J. Uteda

The paper concerns with novel first-order methods for monotone variational inequalities. They use a very simple linesearch procedure that takes into account a local information of the operator. Also the methods do not require…

Optimization and Control · Mathematics 2018-03-26 Yura Malitsky

Inspired by the adaptive Golden Ratio Algorithm (aGRAAL), we propose two new methods for solving monotone variational inequalities. We show that by selecting the momentum parameter beyond the golden ratio in aGRAAL, the convergence speed…

Optimization and Control · Mathematics 2026-04-07 Reza Rahimi Baghbadorani , Peyman Mohajerin Esfahani , Sergio Grammatico

The paper investigates two inertial extragradient algorithms for seeking a common solution to a variational inequality problem involving a monotone and Lipschitz continuous mapping and a fixed point problem with a demicontractive mapping in…

Optimization and Control · Mathematics 2023-08-08 Bing Tan , Liya Liu , Xiaolong Qin

We consider stochastic variational inequality problems where the mapping is monotone over a compact convex set. We present two robust variants of stochastic extragradient algorithms for solving such problems. Of these, the first scheme…

Optimization and Control · Mathematics 2014-03-25 Farzad Yousefian , Angelia Nedic , Uday V. Shanbhag

The article is devoted to the development of numerical methods for solving variational inequalities with relatively strongly monotone operators. We consider two classes of variational inequalities related to some analogs of the Lipschitz…

Optimization and Control · Mathematics 2022-05-25 F. S. Stonyakin , A. A. Titov , D. V. Makarenko , M. S. Alkousa

This paper presents a modified iterative approach to solve the variational inequality problem using the double inertial technique in the context of a real Hilbert space. Our iterative technique involves a projection onto a generalized…

Functional Analysis · Mathematics 2026-03-19 Watanjeet Singh , Sumit Chandok

In this paper, we present two stepsize strategies for the extended Golden Ratio primal-dual algorithm (E-GRPDA) designed to address structured convex optimization problems in finite-dimensional real Hilbert spaces. The first rule features a…

Optimization and Control · Mathematics 2025-12-23 Santanu Soe , Matthew K. Tam , V. Vetrivel

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…

Optimization and Control · Mathematics 2018-03-26 Yu. Malitsky

Golden ratio primal-dual algorithm (GRPDA) is a new variant of the classical Arrow-Hurwicz method for solving structured convex optimization problem, in which the objective function consists of the sum of two closed proper convex functions,…

Optimization and Control · Mathematics 2021-05-18 Xiaokai Chang , Junfeng Yang , Hongchao Zhang

We study monotone variational inequalities that can arise as optimality conditions for constrained convex optimisation or convex-concave minimax problems and propose a novel algorithm that uses only one gradient/operator evaluation and one…

Optimization and Control · Mathematics 2023-07-24 Michael Sedlmayer , Dang-Khoa Nguyen , Radu Ioan Bot

In this work, we propose two step-size strategies for the Golden ratio proximal ADMM (GrpADMM) to solve linearly constrained separable convex optimization problems. Both strategies eliminate explicit operator norm estimates by relying on…

Optimization and Control · Mathematics 2026-04-17 Santanu Soe , V. Vetrivel

In this technical note, we are concerned with the problem of solving variational inequalities with improved convergence rates. Motivated by Nesterov's accelerated gradient method for convex optimization, we propose a Nesterov's accelerated…

Optimization and Control · Mathematics 2022-12-21 Shaolin Tan , Jinhu Lu

Monotone variational inequalities (VIs) provide a unifying framework for convex minimization, equilibrium computation, and convex-concave saddle-point problems. Extragradient-type methods are among the most effective first-order algorithms…

Optimization and Control · Mathematics 2026-04-16 Lingqing Shen , Fatma Kılınç-Karzan

In this article, we provide a modification to the Bregman Golden Ratio Algorithm (B-GRAAL). We analyze the B-GRAAL algorithm with a new step size rule, where the step size increases after a certain number of iterations and does not require…

Optimization and Control · Mathematics 2025-03-11 Gourav Kumar , V. Vetrivel

In this paper we present an inexact zeroth-order method suitable for the solution nonsmooth and nonconvex stochastic composite optimization problems, in which the objective is split into a real-valued Lipschitz continuous stochastic…

Optimization and Control · Mathematics 2025-12-11 Spyridon Pougkakiotis , Dionysis Kalogerias

We propose an extragradient method with stepsizes bounded away from zero for stochastic variational inequalities requiring only pseudo-monotonicity. We provide convergence and complexity analysis, allowing for an unbounded feasible set,…

Optimization and Control · Mathematics 2017-03-02 Alfredo Iusem , Alejandro Jofré , Roberto I. Oliveira , Philip Thompson
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