scholarly journals On the linear convergence of the general first order primal-dual algorithm

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kai Wang ◽  
Deren Han
Keyword(s):  
Linear Convergence ◽  
Analytical Framework ◽  
Dual Algorithm ◽  
Linear Rate ◽  
First Order ◽  
Convex Problems ◽  
Primal Dual ◽  
The One ◽  
Weaker Conditions ◽  
Special Case

<p style='text-indent:20px;'>In this paper, we consider the general first order primal-dual algorithm, which covers several recent popular algorithms such as the one proposed in [Chambolle, A. and Pock T., A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40 (2011) 120-145] as a special case. Under suitable conditions, we prove its global convergence and analyze its linear rate of convergence. As compared to the results in the literature, we derive the corresponding results for the general case and under weaker conditions. Furthermore, the global linear rate of the linearized primal-dual algorithm is established in the same analytical framework.</p>

An improved first-order primal-dual algorithm with a new correction step

2012 ◽  
Vol 57 (4) ◽  
pp. 1419-1428 ◽  
Author(s):  
Xingju Cai ◽  
Deren Han ◽  
Lingling Xu
Keyword(s):  
Dual Algorithm ◽  
First Order ◽  
Primal Dual ◽  
Correction Step

scholarly journals A New Homotopy Proximal Variable-Metric Framework for Composite Convex Minimization

2021 ◽  
Author(s):  
Quoc Tran-Dinh ◽  
Ling Liang ◽  
Kim-Chuan Toh
Keyword(s):  
Convergence Rates ◽  
Optimization Problems ◽  
Linear Convergence ◽  
Convex Minimization ◽  
Variable Metric ◽  
Complexity Bounds ◽  
Convex Problems ◽  
Convex Minimization Problems ◽  
Global Iteration ◽  
Primal Dual

This paper suggests two novel ideas to develop new proximal variable-metric methods for solving a class of composite convex optimization problems. The first idea is to utilize a new parameterization strategy of the optimality condition to design a class of homotopy proximal variable-metric algorithms that can achieve linear convergence and finite global iteration-complexity bounds. We identify at least three subclasses of convex problems in which our approach can apply to achieve linear convergence rates. The second idea is a new primal-dual-primal framework for implementing proximal Newton methods that has attractive computational features for a subclass of nonsmooth composite convex minimization problems. We specialize the proposed algorithm to solve a covariance estimation problem in order to demonstrate its computational advantages. Numerical experiments on the four concrete applications are given to illustrate the theoretical and computational advances of the new methods compared with other state-of-the-art algorithms.

scholarly journals A First-Order Primal-Dual Algorithm with Linesearch

2018 ◽  
Vol 28 (1) ◽  
pp. 411-432 ◽  
Author(s):  
Yura Malitsky ◽  
Thomas Pock
Keyword(s):  
Dual Algorithm ◽  
First Order ◽  
Primal Dual

scholarly journals On the ergodic convergence rates of a first-order primal–dual algorithm

2015 ◽  
Vol 159 (1-2) ◽  
pp. 253-287 ◽  
Author(s):  
Antonin Chambolle ◽  
Thomas Pock
Keyword(s):  
Convergence Rates ◽  
Dual Algorithm ◽  
First Order ◽  
Primal Dual

scholarly journals A First-Order Stochastic Primal-Dual Algorithm with Correction Step

2017 ◽  
Vol 38 (5) ◽  
pp. 602-626 ◽  
Author(s):  
Lorenzo Rosasco ◽  
Silvia Villa ◽  
Bằng Công Vũ
Keyword(s):  
Dual Algorithm ◽  
First Order ◽  
Primal Dual ◽  
Correction Step

scholarly journals Relaxing unimodularity for Yang-Baxter deformed strings

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Stanislav Hronek ◽  
Linus Wulff
Keyword(s):  
Sigma Models ◽  
Bosonic String ◽  
Necessary Condition ◽  
Baxter Equation ◽  
First Order ◽  
Weyl Invariance ◽  
The Matrix ◽  
The One ◽  
Weaker Conditions

Abstract We consider so-called Yang-Baxter deformations of bosonic string sigma- models, based on an R-matrix solving the (modified) classical Yang-Baxter equation. It is known that a unimodularity condition on R is sufficient for Weyl invariance at least to two loops (first order in α′). Here we ask what the necessary condition is. We find that in cases where the matrix (G + B)mn, constructed from the metric and B-field of the undeformed background, is degenerate the unimodularity condition arising at one loop can be replaced by weaker conditions. We further show that for non-unimodular deformations satisfying the one-loop conditions the Weyl invariance extends at least to two loops (first order in α′). The calculations are simplified by working in an O(D, D)-covariant doubled formulation.

scholarly journals An Implementable First-Order Primal-Dual Algorithm for Structured Convex Optimization

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Feng Ma ◽  
Mingfang Ni ◽  
Lei Zhu ◽  
Zhanke Yu
Keyword(s):  
Convex Optimization ◽  
Practical Interest ◽  
Principal Component ◽  
Objective Functions ◽  
Dual Algorithm ◽  
Proximal Point ◽  
Pursuit Problem ◽  
First Order ◽  
Structured Convex Optimization ◽  
Primal Dual

Many application problems of practical interest can be posed as structured convex optimization models. In this paper, we study a new first-order primaldual algorithm. The method can be easily implementable, provided that the resolvent operators of the component objective functions are simple to evaluate. We show that the proposed method can be interpreted as a proximal point algorithm with a customized metric proximal parameter. Convergence property is established under the analytic contraction framework. Finally, we verify the efficiency of the algorithm by solving the stable principal component pursuit problem.

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