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Stochastic Primal-Dual Hybrid Gradient (SPDHG) is an algorithm proposed by Chambolle et al. (2018) to efficiently solve a wide class of nonsmooth large-scale optimization problems. In this paper we contribute to its theoretical foundations…

Optimization and Control · Mathematics 2023-11-27 Eric B Gutierrez , Claire Delplancke , Matthias J Ehrhardt

In this paper, we analyze the recently proposed stochastic primal-dual hybrid gradient (SPDHG) algorithm and provide new theoretical results. In particular, we prove almost sure convergence of the iterates to a solution with convexity and…

Optimization and Control · Mathematics 2022-06-23 Ahmet Alacaoglu , Olivier Fercoq , Volkan Cevher

In this work we propose a new primal-dual algorithm with adaptive step-sizes. The stochastic primal-dual hybrid gradient (SPDHG) algorithm with constant step-sizes has become widely applied in large-scale convex optimization across many…

Optimization and Control · Mathematics 2023-12-05 Antonin Chambolle , Claire Delplancke , Matthias J. Ehrhardt , Carola-Bibiane Schönlieb , Junqi Tang

The linear primal-dual hybrid gradient (PDHG) method is a first-order method that splits convex optimization problems with saddle-point structure into smaller subproblems. Unlike those obtained in most splitting methods, these subproblems…

Optimization and Control · Mathematics 2022-04-05 Jérôme Darbon , Gabriel P. Langlois

We study a block-structured class of convex-concave saddle-point problems in which both the primal and dual variables admit natural separable decompositions. Motivated by large-scale applications where a full update on either side can be…

Optimization and Control · Mathematics 2026-05-19 Yiheng Xiao , Huikang Liu

The primal-dual hybrid gradient (PDHG) algorithm for solving convex optimization problems that arise in tomographic imaging is revisited. In particular, simplification of the selection of step-size parameters is developed for optimization…

Optimization and Control · Mathematics 2026-04-28 Emil Y. Sidky , John Paul Phillips , Zheng Zhang , Dan Xia , Ingrid S. Reiser , Xiaochuan Pan

The Primal-Dual hybrid gradient (PDHG) method is a powerful optimization scheme that breaks complex problems into simple sub-steps. Unfortunately, PDHG methods require the user to choose stepsize parameters, and the speed of convergence is…

Numerical Analysis · Mathematics 2015-03-25 Tom Goldstein , Min Li , Xiaoming Yuan , Ernie Esser , Richard Baraniuk

The primal-dual hybrid gradient (PDHG) algorithm is popular in solving min-max problems which are being widely used in a variety of areas. To improve the applicability and efficiency of PDHG for different application scenarios, we focus on…

Optimization and Control · Mathematics 2023-01-10 Yumin Ma , Xingju Cai , Bo Jiang , Deren Han

In this paper, we propose a stochastic Primal-Dual Hybrid Gradient (PDHG) approach for solving a wide spectrum of regularized stochastic minimization problems, where the regularization term is composite with a linear function. It has been…

Machine Learning · Computer Science 2018-02-02 Linbo Qiao , Tianyi Lin , Qi Qin , Xicheng Lu

We propose an unconstrained optimization method based on the well-known primal-dual hybrid gradient (PDHG) algorithm. We first formulate the optimality condition of the unconstrained optimization problem as a saddle point problem. We then…

Optimization and Control · Mathematics 2024-08-29 X. Zuo , S. Osher , W. Li

We propose a stochastic extension of the primal-dual hybrid gradient algorithm studied by Chambolle and Pock in 2011 to solve saddle point problems that are separable in the dual variable. The analysis is carried out for general…

Optimization and Control · Mathematics 2018-04-11 Antonin Chambolle , Matthias J. Ehrhardt , Peter Richtárik , Carola-Bibiane Schönlieb

Primal-dual algorithms for the resolution of convex-concave saddle point problems usually come with one or several step size parameters. Within the range where convergence is guaranteed, choosing well the step size can make the difference…

Optimization and Control · Mathematics 2024-03-29 Olivier Fercoq

We introduce two novel primal-dual algorithms for addressing nonconvex, nonconcave, and nonsmooth saddle point problems characterized by the weak Minty Variational Inequality (MVI). The first algorithm, Nonconvex-Nonconcave Primal-Dual…

Optimization and Control · Mathematics 2025-06-19 Iyad Walwil , Olivier Fercoq

The primal dual hybrid gradient algorithm (PDHG), which is also known as the Arrow-Hurwicz method, is a fundamental algorithm for saddle point problems especially in imaging. It also inspires a great number of influential algorithms such as…

Optimization and Control · Mathematics 2026-03-03 Shengjie Xu , Bingsheng He

This paper proposes and analyzes a tuning-free variant of Primal-Dual Hybrid Gradient (PDHG), and investigates its effectiveness for solving large-scale semidefinite programming (SDP). The core idea is based on the combination of two…

Optimization and Control · Mathematics 2024-02-02 Yinjun Wang , Haixiang Lan , Yinyu Ye

Convex quadratic programming (QP) is an essential class of optimization problems with broad applications across various fields. Traditional QP solvers, typically based on simplex or barrier methods, face significant scalability challenges.…

Optimization and Control · Mathematics 2024-10-08 Yicheng Huang , Wanyu Zhang , Hongpei Li , Dongdong Ge , Huikang Liu , Yinyu Ye

We propose two variants of the Primal Dual Hybrid Gradient (PDHG) algorithm for saddle point problems with block decomposable duals, hereafter called Multi-Timescale PDHG (MT-PDHG) and its accelerated variant (AMT-PDHG). Through novel…

Optimization and Control · Mathematics 2026-04-03 Junhui Zhang , Patrick Jaillet

In this work we propose a stochastic primal-dual three-operator splitting algorithm (TOS-SPDHG) for solving a class of convex three-composite optimization problems. Our proposed scheme is a direct three-operator splitting extension of the…

Optimization and Control · Mathematics 2025-03-18 Junqi Tang , Matthias Ehrhardt , Carola-Bibiane Schönlieb

The Primal-Dual Hybrid Gradient (PDHG) algorithm is a first-order method that can exploit GPUs to solve large-scale linear programming problems. The approach can often be faster than the alternatives, simplex and interior-point methods,…

Optimization and Control · Mathematics 2026-03-04 Edward Rothberg

Primal-dual hybrid gradient method (PDHG, a.k.a. Chambolle and Pock method) is a well-studied algorithm for minimax optimization problems with a bilinear interaction term. Recently, PDHG is used as the base algorithm for a new LP solver…

Optimization and Control · Mathematics 2023-03-14 Haihao Lu , Jinwen Yang
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