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High-efciency improved symmetric successive over-relaxation preconditioned conjugate gradient method for solving large-scale finite element linear equations

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Applied Mathematics and Mechanics(English Edition) 2013年第10期34卷 1225-1236页

作者：李根唐春安李连崇 Department of Applied Mechanics and Engineering School of EngineeringSun Yat-sen University Institute for Rock Instability and Seismicity Research School of Civil EngineeringDalian University of Technology

Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering （CAE） and computer aided manufacturing （CAM）. This paper presents a high-efficiency improved symmetric successive over-relaxation （ISSOR） preconditioned conjugate gradient （PCG） method, which maintains lelism consistent with the original form. Ideally, the by 50% as compared with the original algorithm. the convergence and inherent paralcomputation can It is suitable for be reduced nearly high-performance computing with its inherent basic high-efficiency operations. By comparing with the numerical results, it is shown that the proposed method has the best performance.

关键词： improved preconditioned conjugate gradient (PCG) method conjugate gradient method large-scale linear equation finite element method

A Barzilai-Borwein conjugate gradient method

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Science China Mathematics 2016年第8期59卷 1511-1524页

作者： DAI YuHong KOU CaiXia LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems Science Chinese Academy of Sciences School of Science Beijing University of Posts and Telecommunications

The linear conjugate gradient method is an optimal method for convex quadratic minimization due to the Krylov subspace minimization property. The proposition of limited-memory BFGS method and Barzilai-Borwein gradient method, however, heavily restricted the use of conjugate gradient method for largescale nonlinear optimization. This is, to the great extent, due to the requirement of a relatively exact line search at each iteration and the loss of conjugacy property of the search directions in various occasions. On the contrary, the limited-memory BFGS method and the Barzilai-Bowein gradient method share the so-called asymptotical one stepsize per line-search property, namely, the trial stepsize in the method will asymptotically be accepted by the line search when the iteration is close to the solution. This paper will focus on the analysis of the subspace minimization conjugate gradient method by Yuan and Stoer(1995). Specifically, if choosing the parameter in the method by combining the Barzilai-Borwein idea, we will be able to provide some efficient Barzilai-Borwein conjugate gradient(BBCG) methods. The initial numerical experiments show that one of the variants, BBCG3, is specially efficient among many others without line searches. This variant of the BBCG method might enjoy the asymptotical one stepsize per line-search property and become a strong candidate for large-scale nonlinear optimization.

关键词： conjugate gradient method subspace minimization Barzilai-Bowein gradient method line search descent property global convergence

A modified three–term conjugate gradient method with sufficient descent property

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Applied Mathematics(A Journal of Chinese Universities) 2015年第3期30卷 263-272页

作者： Saman Babaie–Kafaki Department of Mathematics Faculty of Mathematics Statistics and Computer Science Semnan University

A hybridization of the three–term conjugate gradient method proposed by Zhang et al. and the nonlinear conjugate gradient method proposed by Polak and Ribi`ere, and Polyak is suggested. Based on an eigenvalue analysis, it is shown that search directions of the proposed method satisfy the sufficient descent condition, independent of the line search and the objective function convexity. Global convergence of the method is established under an Armijo–type line search condition. Numerical experiments show practical efficiency of the proposed method.

关键词： unconstrained optimization conjugate gradient method eigenvalue sufficient descent condition global convergence

A REGULARIZED conjugate gradient method FOR SYMMETRIC POSITIVE DEFINITE SYSTEM OF LINEAR EQUATIONS

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Journal of Computational Mathematics 2002年第4期20卷 437-448页

作者： Zhong-zhi Bai Shao-liang Zhang LSEC ICMSECAcademy of Mathematics and System SciencesChinese Academy of SciencesBeijing 100080China Department of Applied Physics University of Tokyo7-3-1 HongoBunkyo-kuTokyo 113-8656Japan

A class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The convergence properties of these methods are discussed in depth, and the best possible choices of the parameters involved in the new methods are investigated in detail. Numerical computations show that the new methods are more efficient and robust than both classical relaxation methods and classical conjugate direction methods.

关键词： conjugate gradient method symmetric positive definite matrix regularization ill-conditioned linear system

A Preconditioned conjugate gradient method with Active Set Strategy for1-Regularized Least Squares

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Journal of the Operations Research Society of China 2018年第4期6卷 571-585页

作者： Wan-You Cheng Dong-Hui Li College of Computer Dongguan University of TechnologyDongguan 523000GuangdongChina School of Mathematical Sciences South China Normal UniversityGuangzhou 510631China

In the paper,we consider the l_(1)-regularized least square problem which has been intensively involved in the fields of signal processing,compressive sensing,linear inverse problems and statistical *** considered problem has been proved recently to be equivalent to a nonnegatively constrained quadratic programming(QP).In this paper,we use a recently developed active conjugate gradient method to solve the resulting QP *** improve the algorithm’s performance,we design a subspace exact steplength as well as a precondition *** performance comparisons illustrate that the proposed algorithm is competitive and even performs little better than several state-of-the-art algorithms.

关键词： Compressed sensing l_(1)-Regularized optimization conjugate gradient method Precondition

Subspace Minimization conjugate gradient method Based on Cubic Regularization Model for Unconstrained Optimization

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Journal of Harbin Institute of Technology(New Series) 2021年第5期28卷 61-69页

作者： Ting Zhao Hongwei Liu School of Mathematics and Statistics Xidian UniversityXi'an 710126China

Many methods have been put forward to solve unconstrained optimization problems,among which conjugate gradient method(CG)is very *** the increasing emergence of large⁃scale problems,the subspace technology has become particularly important and widely used in the field of *** this study,a new CG method was put forward,which combined subspace technology and a cubic regularization ***,a special scaled norm in a cubic regularization model was *** certain conditions,some significant characteristics of the search direction were given and the convergence of the algorithm was *** comparisons show that for the 145 test functions under the CUTEr library,the proposed method is better than two classical CG methods and two new subspaces conjugate gradient methods.

关键词： cubic regularization model conjugate gradient method subspace technique unconstrained optimization

Global Convergence of a Modified Spectral CD conjugate gradient method

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Journal of Mathematical Research and Exposition 2011年第2期31卷 261-268页

作者： Wei CAO Kai Rong WANG Yi Li WANG College of Mathematics and Statistics Chongqing University Chongqing 401331 P. R. China

In this paper,we present a new nonlinear modified spectral CD conjugate gradient method for solving large scale unconstrained optimization *** direction generated by the method is a descent direction for the objective function,and this property depends neither on the line search rule,nor on the convexity of the objective ***,the modified method reduces to the standard CD method if line search is *** some mild conditions,we prove that the modified method with line search is globally convergent even if the objective function is *** numerical results show that the proposed method is very promising.

关键词： unconstrained optimization conjugate gradient method armijo-type line search global convergence

A Modified Hestenes-Stiefel conjugate gradient method and Its Convergence

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Journal of Mathematical Research and Exposition 2010年第2期30卷 297-308页

作者： Zeng Xin WEI Hai Dong HUANG Yah Rong TAO College of Mathematics and Information Science Guangxi University Guangxi 530004 P. R. China

It is well-known that the direction generated by Hestenes-Stiefel （HS） conjugate gradient method may not be a descent direction for the objective function. In this paper, we take a little modification to the HS method, then the generated direction always satisfies the sufficient descent condition. An advantage of the modified Hestenes-Stiefel （MHS） method is that the scalar βκHS. keeps nonnegative under the weak Wolfe-Powell line search. The global convergence result of the MHS method is established under some mild conditions. Preliminary numerical results show that the MHS method is a little more efficient than PRP and HS methods.

关键词： conjugate gradient method sufficient descent condition line search global convergence.

Global Convergence of a Modified PRP conjugate gradient method

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Journal of Mathematical Research and Exposition 2010年第1期30卷 141-148页

作者： Hai Dong HUANG Yan Jun LI Zeng Xin WEI College of Mathematics and Information Science Guangxi University Guangxi 530004 P. R. China

In this paper, a modified formula for βk^PRP is proposed for the conjugate gradient method of solving unconstrained optimization problems. The value of βk^PRP keeps nonnegative independent of the line search. Under mild conditions, the global convergence of modified PRP method with the strong Wolfe-Powell line search is established. Preliminary numerical results show that the modified method is efficient.

关键词： unconstrained optimization conjugate gradient method global convergence.