近年来,PRSM成为了处理具有线性约束的两部分可分凸优化问题的一个热门研究方向。本研究聚焦于目标函数由三个解耦变量函数之和构成的可分凸优化问题。单纯地运用PRSM可能无法保证其收敛性。因此,我们引入了一种带有惯性项和临近项的严...近年来,PRSM成为了处理具有线性约束的两部分可分凸优化问题的一个热门研究方向。本研究聚焦于目标函数由三个解耦变量函数之和构成的可分凸优化问题。单纯地运用PRSM可能无法保证其收敛性。因此,我们引入了一种带有惯性项和临近项的严格收缩PRSM。借助变分不等式、临近点算法以及基本不等式,我们对所提出的方法进行了全局收敛性的分析。此外,我们将这种新方法应用于鲁棒主成分分析(PCA)问题的求解,并提供了一些初步的数值结果,用以展示该方法的可行性和有效性。In recent years, PRSM has become a popular research direction for dealing with two-block separable convex optimization problems with linear constraints. This study focuses on separable convex optimization problems where the objective function is composed of the sum of three decoupled variable functions. Simply applying the PRSM may not guarantee its convergence. Therefore, we introduce a strictly contractive PRSM with inertial and proximal terms. By means of variational inequalities, the proximal point algorithm, and fundamental inequalities, we have analyzed the global convergence of the proposed method. In addition, we applied this new method to the solution of the Robust Principal Component Analysis (PCA) problem and provided some preliminary numerical results to demonstrate the feasibility and effectiveness of the method.展开更多
The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints.However,its convergence was not guaranteed without extra requ...The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints.However,its convergence was not guaranteed without extra requirements.He et al.(SIAM J.Optim.24:1011-1040,2014)proved the convergence of a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation factor.In this paper,we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation factors.Besides,motivated by the recent advances on the ADMM type method with indefinite proximal terms,we employ the indefinite proximal term in the strictly contractive Peaceman-Rachford splitting method.We show that the proposed indefinite-proximal strictly contractive Peaceman-Rachford splitting method is convergent and also prove the o(1/t)convergence rate in the nonergodic sense.The numerical tests on the l 1 regularized least square problem demonstrate the efficiency of the proposed method.展开更多
The strictly contractive Peaceman-Rachford splitting method is one of effective methods for solving separable convex optimization problem, and the inertial proximal Peaceman-Rachford splitting method is one of its imp...The strictly contractive Peaceman-Rachford splitting method is one of effective methods for solving separable convex optimization problem, and the inertial proximal Peaceman-Rachford splitting method is one of its important variants. It is known that the convergence of the inertial proximal Peaceman- Rachford splitting method can be ensured if the relaxation factor in Lagrangian multiplier updates is underdetermined, which means that the steps for the Lagrangian multiplier updates are shrunk conservatively. Although small steps play an important role in ensuring convergence, they should be strongly avoided in practice. In this article, we propose a relaxed inertial proximal Peaceman- Rachford splitting method, which has a larger feasible set for the relaxation factor. Thus, our method provides the possibility to admit larger steps in the Lagrangian multiplier updates. We establish the global convergence of the proposed algorithm under the same conditions as the inertial proximal Peaceman-Rachford splitting method. Numerical experimental results on a sparse signal recovery problem in compressive sensing and a total variation based image denoising problem demonstrate the effectiveness of our method.展开更多
This work is about a splitting method for solving a nonconvex nonseparable optimization problem with linear constraints,where the objective function consists of two separable functions and a coupled term.First,based o...This work is about a splitting method for solving a nonconvex nonseparable optimization problem with linear constraints,where the objective function consists of two separable functions and a coupled term.First,based on the ideas from Bregman distance and Peaceman–Rachford splitting method,the Bregman Peaceman–Rachford splitting method with different relaxation factors for the multiplier is proposed.Second,the global and strong convergence of the proposed algorithm are proved under general conditions including the region of the two relaxation factors as well as the crucial Kurdyka–Łojasiewicz property.Third,when the associated Kurdyka–Łojasiewicz property function has a special structure,the sublinear and linear convergence rates of the proposed algorithm are guaranteed.Furthermore,some preliminary numerical results are shown to indicate the effectiveness of the proposed algorithm.展开更多
This work is devoted to studying an accelerated stochastic Peaceman–Rachford splitting method(AS-PRSM)for solving a family of structural empirical risk minimization problems.The objective function to be optimized is ...This work is devoted to studying an accelerated stochastic Peaceman–Rachford splitting method(AS-PRSM)for solving a family of structural empirical risk minimization problems.The objective function to be optimized is the sum of a possibly nonsmooth convex function and a finite sum of smooth convex component functions.The smooth subproblem in AS-PRSM is solved by a stochastic gradient method using variance reduction technique and accelerated techniques,while the possibly nonsmooth subproblem is solved by introducing an indefinite proximal term to transform its solution into a proximity operator.By a proper choice for the involved parameters,we show that AS-PRSM converges in a sublinear convergence rate measured by the function value residual and constraint violation in the sense of expectation and ergodic.Preliminary experiments on testing the popular graph-guided fused lasso problem in machine learning and the 3D CT reconstruction problem in medical image processing show that the proposed AS-PRSM is very efficient.展开更多
文摘近年来,PRSM成为了处理具有线性约束的两部分可分凸优化问题的一个热门研究方向。本研究聚焦于目标函数由三个解耦变量函数之和构成的可分凸优化问题。单纯地运用PRSM可能无法保证其收敛性。因此,我们引入了一种带有惯性项和临近项的严格收缩PRSM。借助变分不等式、临近点算法以及基本不等式,我们对所提出的方法进行了全局收敛性的分析。此外,我们将这种新方法应用于鲁棒主成分分析(PCA)问题的求解,并提供了一些初步的数值结果,用以展示该方法的可行性和有效性。In recent years, PRSM has become a popular research direction for dealing with two-block separable convex optimization problems with linear constraints. This study focuses on separable convex optimization problems where the objective function is composed of the sum of three decoupled variable functions. Simply applying the PRSM may not guarantee its convergence. Therefore, we introduce a strictly contractive PRSM with inertial and proximal terms. By means of variational inequalities, the proximal point algorithm, and fundamental inequalities, we have analyzed the global convergence of the proposed method. In addition, we applied this new method to the solution of the Robust Principal Component Analysis (PCA) problem and provided some preliminary numerical results to demonstrate the feasibility and effectiveness of the method.
基金supported by the Natural Science Foundation of Jiangsu Province (Grant No.BK20210267)supported by the National Natural Science Foundation of China (Grant No.11971239)+1 种基金the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (Grant No.21KJA110002)supported by the National Natural Science Foundation of China (Grant Nos.12131004,11625105).
文摘The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints.However,its convergence was not guaranteed without extra requirements.He et al.(SIAM J.Optim.24:1011-1040,2014)proved the convergence of a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation factor.In this paper,we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation factors.Besides,motivated by the recent advances on the ADMM type method with indefinite proximal terms,we employ the indefinite proximal term in the strictly contractive Peaceman-Rachford splitting method.We show that the proposed indefinite-proximal strictly contractive Peaceman-Rachford splitting method is convergent and also prove the o(1/t)convergence rate in the nonergodic sense.The numerical tests on the l 1 regularized least square problem demonstrate the efficiency of the proposed method.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant Nos. 11671116, 11271107, 91630202） and the Natural Science Foundation of Hebei Province of China （No. A2015202365）.
文摘The strictly contractive Peaceman-Rachford splitting method is one of effective methods for solving separable convex optimization problem, and the inertial proximal Peaceman-Rachford splitting method is one of its important variants. It is known that the convergence of the inertial proximal Peaceman- Rachford splitting method can be ensured if the relaxation factor in Lagrangian multiplier updates is underdetermined, which means that the steps for the Lagrangian multiplier updates are shrunk conservatively. Although small steps play an important role in ensuring convergence, they should be strongly avoided in practice. In this article, we propose a relaxed inertial proximal Peaceman- Rachford splitting method, which has a larger feasible set for the relaxation factor. Thus, our method provides the possibility to admit larger steps in the Lagrangian multiplier updates. We establish the global convergence of the proposed algorithm under the same conditions as the inertial proximal Peaceman-Rachford splitting method. Numerical experimental results on a sparse signal recovery problem in compressive sensing and a total variation based image denoising problem demonstrate the effectiveness of our method.
基金supported by the National Natural Science Foundation of China(No.12171106)the Natural Science Foundation of Guangxi Province(Nos.2020GXNSFDA238017 and 2018GXNSFFA281007).
文摘This work is about a splitting method for solving a nonconvex nonseparable optimization problem with linear constraints,where the objective function consists of two separable functions and a coupled term.First,based on the ideas from Bregman distance and Peaceman–Rachford splitting method,the Bregman Peaceman–Rachford splitting method with different relaxation factors for the multiplier is proposed.Second,the global and strong convergence of the proposed algorithm are proved under general conditions including the region of the two relaxation factors as well as the crucial Kurdyka–Łojasiewicz property.Third,when the associated Kurdyka–Łojasiewicz property function has a special structure,the sublinear and linear convergence rates of the proposed algorithm are guaranteed.Furthermore,some preliminary numerical results are shown to indicate the effectiveness of the proposed algorithm.
基金supported by the National Natural Science Foundation of China(Nos.12001430,11972292 and 12161053)the China Postdoctoral Science Foundation(No.2020M683545)the Guangdong Basic and Applied Basic Research Foundation(No.2023A1515012405).
文摘This work is devoted to studying an accelerated stochastic Peaceman–Rachford splitting method(AS-PRSM)for solving a family of structural empirical risk minimization problems.The objective function to be optimized is the sum of a possibly nonsmooth convex function and a finite sum of smooth convex component functions.The smooth subproblem in AS-PRSM is solved by a stochastic gradient method using variance reduction technique and accelerated techniques,while the possibly nonsmooth subproblem is solved by introducing an indefinite proximal term to transform its solution into a proximity operator.By a proper choice for the involved parameters,we show that AS-PRSM converges in a sublinear convergence rate measured by the function value residual and constraint violation in the sense of expectation and ergodic.Preliminary experiments on testing the popular graph-guided fused lasso problem in machine learning and the 3D CT reconstruction problem in medical image processing show that the proposed AS-PRSM is very efficient.
