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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

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

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…

Optimization and Control · Mathematics 2019-05-27 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

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

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

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 paper, we focus on the problem of minimizing a continuously differentiable convex objective function, $\min_x f(x)$. Recently, Malitsky (2020); Alacaoglu et al.(2023) developed an adaptive first-order method, GRAAL. This algorithm…

Optimization and Control · Mathematics 2025-09-01 Ekaterina Borodich , Dmitry Kovalev

In this paper, we propose an inexact golden ratio primal-dual algorithm with linesearch step(IP-GRPDAL) for solving the saddle point problems, where two subproblems can be approximately solved by applying the notations of inexact extended…

Optimization and Control · Mathematics 2025-09-23 Changjie Fang , Jinxiu Liu , Jingtao Qiu , Shenglan Chen

In this paper, we propose some accelerated methods for solving optimization problems under the condition of relatively smooth and relatively Lipschitz continuous functions with an inexact oracle. We consider the problem of minimizing the…

Optimization and Control · Mathematics 2024-11-27 O. S. Savchuk , M. S. Alkousa , A. S. Shushko , A. A. Vyguzov , F. S. Stonyakin , D. A. Pasechnyuk , A. V. Gasnikov

We consider the problem of minimizing the sum of a Lipschitz differentiable convex function $f$ and a proper closed convex function $h$ that admits efficient linear minimization oracles, subject to multiple smooth convex inequality…

Optimization and Control · Mathematics 2026-05-22 Xiaozhou Wang , Ting Kei Pong , Zev Woodstock

Stochastic variance reduced methods have shown strong performance in solving finite-sum problems. However, these methods usually require the users to manually tune the step-size, which is time-consuming or even infeasible for some…

Optimization and Control · Mathematics 2023-10-10 Binghui Xie , Chenhan Jin , Kaiwen Zhou , James Cheng , Wei Meng

Many problems in machine learning write as the minimization of a sum of individual loss functions over the training examples. These functions are usually differentiable but, in some cases, their gradients are not Lipschitz continuous, which…

Optimization and Control · Mathematics 2024-04-29 S. Chraibi , F. Iutzeler , J. Malick , A. Rogozin

We propose a novel stochastic distributed method for both monotone and strongly monotone variational inequalities with Lipschitz operator and proper convex regularizers arising in various applications from game theory to adversarial…

Optimization and Control · Mathematics 2024-10-07 Aleksandr Beznosikov , Darina Dvinskikh , Dmitry Bylinkin , Andrei Semenov , Alexander Gasnikov

Selecting an effective step-size is a fundamental challenge in first-order optimization, especially for problems with non-Euclidean geometries. This paper presents a novel adaptive step-size strategy for optimization algorithms that rely on…

Optimization and Control · Mathematics 2025-10-14 Abbas Khademi , Antonio Silveti-Falls

The main purpose of this paper is to propose a variance-based Bregman extragradient algorithm with line search for solving stochastic variational inequalities, which is robust with respect an unknown Lipschitz constant. We prove the almost…

Optimization and Control · Mathematics 2022-08-31 Xian-Jun Long , Yue-Hong He , Nan-Jing Huang

This paper proposes a novel approach to adaptive step sizes in stochastic gradient descent (SGD) by utilizing quantities that we have identified as numerically traceable -- the Lipschitz constant for gradients and a concept of the local…

Optimization and Control · Mathematics 2024-09-19 Frederik Köhne , Leonie Kreis , Anton Schiela , Roland Herzog

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

This paper introduces adaptive Bregman proximal gradient algorithms for solving convex composite minimization problems without relying on global relative smoothness or strong convexity assumptions. Building upon recent advances in adaptive…

Optimization and Control · Mathematics 2025-08-05 Hongjia Ou , Puya Latafat , Andreas Themelis

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
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