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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

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

The primal-dual hybrid gradient method (PDHG) is useful for optimization problems that commonly appear in image reconstruction. A downside of PDHG is that there are typically three user-set parameters and performance of the algorithm is…

Optimization and Control · Mathematics 2025-03-25 Alex McManus , Stephen Becker , Nicholas Dwork

The Stochastic Primal-Dual Hybrid Gradient (SPDHG) was proposed by Chambolle et al. (2018) and is an efficient algorithm to solve some nonsmooth large-scale optimization problems. In this paper we prove its almost sure convergence for…

Optimization and Control · Mathematics 2021-04-02 Eric B. Gutierrez , Claire Delplancke , Matthias J. Ehrhardt

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

Image restoration remains a challenging task in image processing. Numerous methods tackle this problem, often solved by minimizing a non-smooth penalized co-log-likelihood function. Although the solution is easily interpretable with…

Computer Vision and Pattern Recognition · Computer Science 2021-12-21 Mingyuan Jiu , Nelly Pustelnik

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

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

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

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 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

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

A wide array of image recovery problems can be abstracted into the problem of minimizing a sum of composite convex functions in a Hilbert space. To solve such problems, primal-dual proximal approaches have been developed which provide…

Optimization and Control · Mathematics 2014-06-23 Patrick L. Combettes , Laurent Condat , Jean-Christophe Pesquet , Bang Cong Vu

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

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

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 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

The primal-dual hybrid gradient (PDHG) method is one of the most popular algorithms for solving saddle point problems. However, when applying the PDHG method and its many variants to some real-world models commonly encountered in signal…

Optimization and Control · Mathematics 2025-06-10 Jintao Yu , Hongjin He
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