In this paper,we propose a three-term conjugate gradient method for solving unconstrained optimization problems based on the Hestenes-Stiefel(HS)conjugate gradient method and Polak-Ribiere-Polyak(PRP)conjugate gradien...In this paper,we propose a three-term conjugate gradient method for solving unconstrained optimization problems based on the Hestenes-Stiefel(HS)conjugate gradient method and Polak-Ribiere-Polyak(PRP)conjugate gradient method.Under the condition of standard Wolfe line search,the proposed search direction is the descent direction.For general nonlinear functions,the method is globally convergent.Finally,numerical results show that the proposed method is efficient.展开更多
In this paper, we propose a globally convergent Polak-Ribiere-Polyak (PRP) conjugate gradient method for nonconvex minimization of differentiable functions by employing an Armijo-type line search which is simpler and ...In this paper, we propose a globally convergent Polak-Ribiere-Polyak (PRP) conjugate gradient method for nonconvex minimization of differentiable functions by employing an Armijo-type line search which is simpler and less demanding than those defined in [4,10]. A favorite property of this method is that we can choose the initial stepsize as the one-dimensional minimizer of a quadratic modelΦ(t):= f(xk)+tgkTdk+(1/2) t2dkTQkdk, where Qk is a positive definite matrix that carries some second order information of the objective function f. So, this line search may make the stepsize tk more easily accepted. Preliminary numerical results show that this method is efficient.展开更多
This paper puts forward a two-parameter family of nonlinear conjugate gradient(CG)method without line search for solving unconstrained optimization problem.The main feature of this method is that it does not rely on a...This paper puts forward a two-parameter family of nonlinear conjugate gradient(CG)method without line search for solving unconstrained optimization problem.The main feature of this method is that it does not rely on any line search and only requires a simple step size formula to always generate a sufficient descent direction.Under certain assumptions,the proposed method is proved to possess global convergence.Finally,our method is compared with other potential methods.A large number of numerical experiments show that our method is more competitive and effective.展开更多
As a generalization of the two-term conjugate gradient method(CGM),the spectral CGM is one of the effective methods for solving unconstrained optimization.In this paper,we enhance the JJSL conjugate parameter,initiall...As a generalization of the two-term conjugate gradient method(CGM),the spectral CGM is one of the effective methods for solving unconstrained optimization.In this paper,we enhance the JJSL conjugate parameter,initially proposed by Jiang et al.(Computational and Applied Mathematics,2021,40:174),through the utilization of a convex combination technique.And this improvement allows for an adaptive search direction by integrating a newly constructed spectral gradient-type restart strategy.Then,we develop a new spectral CGM by employing an inexact line search to determine the step size.With the application of the weak Wolfe line search,we establish the sufficient descent property of the proposed search direction.Moreover,under general assumptions,including the employment of the strong Wolfe line search for step size calculation,we demonstrate the global convergence of our new algorithm.Finally,the given unconstrained optimization test results show that the new algorithm is effective.展开更多
Recently, Gilbert and Nocedal([3]) investigated global convergence of conjugate gradient methods related to Polak-Ribiere formular, they restricted beta(k) to non-negative value. [5] discussed the same problem as that...Recently, Gilbert and Nocedal([3]) investigated global convergence of conjugate gradient methods related to Polak-Ribiere formular, they restricted beta(k) to non-negative value. [5] discussed the same problem as that in [3] and relaxed beta(k) to be negative with the objective function being convex. This paper allows beta(k) to be selected in a wider range than [5]. Especially, the global convergence of the corresponding algorithm without sufficient decrease condition is proved.展开更多
In this paper, we present a new hybrid conjugate gradient algorithm for unconstrained optimization. This method is a convex combination of Liu-Storey conjugate gradient method and Fletcher-Reeves conjugate gradient me...In this paper, we present a new hybrid conjugate gradient algorithm for unconstrained optimization. This method is a convex combination of Liu-Storey conjugate gradient method and Fletcher-Reeves conjugate gradient method. We also prove that the search direction of any hybrid conjugate gradient method, which is a convex combination of two conjugate gradient methods, satisfies the famous D-L conjugacy condition and in the same time accords with the Newton direction with the suitable condition. Furthermore, this property doesn't depend on any line search. Next, we also prove that, moduling the value of the parameter t,the Newton direction condition is equivalent to Dai-Liao conjugacy condition.The strong Wolfe line search conditions are used.The global convergence of this new method is proved.Numerical comparisons show that the present hybrid conjugate gradient algorithm is the efficient one.展开更多
With the development of computational power, there has been an increased focus on data-fitting related seismic inversion techniques for high fidelity seismic velocity model and image, such as full-waveform inversion a...With the development of computational power, there has been an increased focus on data-fitting related seismic inversion techniques for high fidelity seismic velocity model and image, such as full-waveform inversion and least squares migration. However, though more advanced than conventional methods, these data fitting methods can be very expensive in terms of computational cost. Recently, various techniques to optimize these data-fitting seismic inversion problems have been implemented to cater for the industrial need for much improved efficiency. In this study, we propose a general stochastic conjugate gradient method for these data-fitting related inverse problems. We first prescribe the basic theory of our method and then give synthetic examples. Our numerical experiments illustrate the potential of this method for large-size seismic inversion application.展开更多
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 ...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.展开更多
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 analysi...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.展开更多
In one step inverse finite element approach, an initial blank shape is normally predicted from the final deformed shape. The final deformed shape needs to be trimmed into a final part after stamping, the trimmed area,...In one step inverse finite element approach, an initial blank shape is normally predicted from the final deformed shape. The final deformed shape needs to be trimmed into a final part after stamping, the trimmed area, therefore, needs to be compensated manually before using one step inverse approach, which causes low efficiency and in consistency with the real situation. To solve this problem, one step positive approach is proposed to simulate the sheet metal stamping process. Firstly the spatial initial solution of one step positive method is preliminarily obtained by using the mapping relationship and area coordinates, then based on the deformation theory the iterative solving is carried out in three-dimensional coordinate system by using quasi-conjugate-gradient method. During iterative process the contact judgment method is introduced to ensure that the nodes on the spatial initial solution are not separated from die surface. The predicted results of sheet metal forming process that include the shape and thickness of the stamped part can be obtained after the iterative solving process. The validity of the proposed approach is verified by comparing the predicted results obtained through the proposed approach with those obtained through the module of one step inverse approach in Autoform and the real stamped part. In one step positive method, the stamped shape of regular sheet can be calculated fast and effectively. During the iterative solution, the quasi-conjugate-gradient method is proposed to take the place of solving system of equations, and it can improve the stability and precision of the algorithm.展开更多
Many methods have been put forward to solve unconstrained optimization problems,among which conjugate gradient method(CG)is very important.With the increasing emergence of large⁃scale problems,the subspace technology ...Many methods have been put forward to solve unconstrained optimization problems,among which conjugate gradient method(CG)is very important.With the increasing emergence of large⁃scale problems,the subspace technology has become particularly important and widely used in the field of optimization.In this study,a new CG method was put forward,which combined subspace technology and a cubic regularization model.Besides,a special scaled norm in a cubic regularization model was analyzed.Under certain conditions,some significant characteristics of the search direction were given and the convergence of the algorithm was built.Numerical 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.展开更多
Conjugate gradient optimization algorithms depend on the search directions with different choices for the parameters in the search directions. In this note, by combining the nice numerical performance of PR and HS met...Conjugate gradient optimization algorithms depend on the search directions with different choices for the parameters in the search directions. In this note, by combining the nice numerical performance of PR and HS methods with the global convergence property of the class of conjugate gradient methods presented by HU and STOREY(1991), a class of new restarting conjugate gradient methods is presented. Global convergences of the new method with two kinds of common line searches, are proved. Firstly, it is shown that, using reverse modulus of continuity function and forcing function, the new method for solving unconstrained optimization can work for a continously dif ferentiable function with Curry-Altman's step size rule and a bounded level set. Secondly, by using comparing technique, some general convergence properties of the new method with other kind of step size rule are established. Numerical experiments show that the new method is efficient by comparing with FR conjugate gradient method.展开更多
In this paper,an efficient conjugate gradient method is given to solve the general unconstrained optimization problems,which can guarantee the sufficient descent property and the global convergence with the strong Wol...In this paper,an efficient conjugate gradient method is given to solve the general unconstrained optimization problems,which can guarantee the sufficient descent property and the global convergence with the strong Wolfe line search conditions.Numerical results show that the new method is efficient and stationary by comparing with PRP＋ method,so it can be widely used in scientific computation.展开更多
In this note,by combining the nice numerical performance of PR and HS methods with the global convergence property of FR method,a class of new restarting three terms conjugate gradient methods is presented.Global conv...In this note,by combining the nice numerical performance of PR and HS methods with the global convergence property of FR method,a class of new restarting three terms conjugate gradient methods is presented.Global convergence properties of the new method with two kinds of common line searches are proved.展开更多
Nonlinear conjugate gradient methods have played an important role in solving large scale unconstrained optimi-zation problems,it is characterized by the simplicity of their iteration and their low memory requirements...Nonlinear conjugate gradient methods have played an important role in solving large scale unconstrained optimi-zation problems,it is characterized by the simplicity of their iteration and their low memory requirements.It is well-known that the direction generated by a conjugate gradient method may be not a descent direction.In this paper,a new class of nonlinear conjugate gradient method is presented,its search direction is a descent direction for the objective function.If the objective function is differentiable and its gradient is Lipschitz continuous,the line sbarch satisfies strong Wolfe condition,the global convergence result is established.展开更多
In this paper, a class of the stochastic generalized linear complementarity problems with finitely many elements is proposed for the first time. Based on the Fischer-Burmeister function, a new conjugate gradient proje...In this paper, a class of the stochastic generalized linear complementarity problems with finitely many elements is proposed for the first time. Based on the Fischer-Burmeister function, a new conjugate gradient projection method is given for solving the stochastic generalized linear complementarity problems. The global convergence of the conjugate gradient projection method is proved and the related numerical results are also reported.展开更多
In this paper we consider the global convergence of any conjugate gradient method of the form d1=-g1,dk+1=-gk+1+βkdk(k≥1)with any βk satisfying sume conditions,and with the strong wolfe line search conditions.Under...In this paper we consider the global convergence of any conjugate gradient method of the form d1=-g1,dk+1=-gk+1+βkdk(k≥1)with any βk satisfying sume conditions,and with the strong wolfe line search conditions.Under the convex assumption on the objective function,we preve the descenf property and the global convergence of this method.展开更多
In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method (PCCG). The theorems discuss respectively the qualitative property of the ite...In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method (PCCG). The theorems discuss respectively the qualitative property of the iterative solution and the construction principle of the iterative matrix. The authors put forward a new incompletely LU factorizing technique for non-M-matrix and the method of constructing the iterative matrix. This improved PCCG is used to calculate the ill-conditioned problems and large-scale three-dimensional finite element problems, and simultaneously contrasted with other methods. The abnormal phenomenon is analyzed when PCCG is used to solve the system of ill-conditioned equations, ft is shown that the method proposed in this paper is quite effective in solving the system of large-scale finite element equations and the system of ill-conditioned equations.展开更多
In this paper, we propose a method to solve coupled problem. Our computational method is mainly based on conjugate gradient algorithm. We use finite difference method for the structure and finite element method for th...In this paper, we propose a method to solve coupled problem. Our computational method is mainly based on conjugate gradient algorithm. We use finite difference method for the structure and finite element method for the fluid. Conjugate gradient method gives suitable numerical results according to some papers.展开更多
基金Supported by the Science and Technology Project of Guangxi(Guike AD23023002)。
文摘In this paper,we propose a three-term conjugate gradient method for solving unconstrained optimization problems based on the Hestenes-Stiefel(HS)conjugate gradient method and Polak-Ribiere-Polyak(PRP)conjugate gradient method.Under the condition of standard Wolfe line search,the proposed search direction is the descent direction.For general nonlinear functions,the method is globally convergent.Finally,numerical results show that the proposed method is efficient.
基金This work is supported by the Chinese NSF grants 60475042 Guangxi NSF grants 0542043the Foundation of Advanced Research Center of Zhongshan University and Hong Kong
文摘In this paper, we propose a globally convergent Polak-Ribiere-Polyak (PRP) conjugate gradient method for nonconvex minimization of differentiable functions by employing an Armijo-type line search which is simpler and less demanding than those defined in [4,10]. A favorite property of this method is that we can choose the initial stepsize as the one-dimensional minimizer of a quadratic modelΦ(t):= f(xk)+tgkTdk+(1/2) t2dkTQkdk, where Qk is a positive definite matrix that carries some second order information of the objective function f. So, this line search may make the stepsize tk more easily accepted. Preliminary numerical results show that this method is efficient.
基金Supported by 2023 Inner Mongolia University of Finance and Economics,General Scientific Research for Universities directly under Inner Mon‐golia,China (NCYWT23026)2024 High-quality Research Achievements Cultivation Fund Project of Inner Mongolia University of Finance and Economics,China (GZCG2479)。
文摘This paper puts forward a two-parameter family of nonlinear conjugate gradient(CG)method without line search for solving unconstrained optimization problem.The main feature of this method is that it does not rely on any line search and only requires a simple step size formula to always generate a sufficient descent direction.Under certain assumptions,the proposed method is proved to possess global convergence.Finally,our method is compared with other potential methods.A large number of numerical experiments show that our method is more competitive and effective.
基金supported by the National Natural Science Foundation of China(No.72071202)the Key Laboratory of Mathematics and Engineering Applications,Ministry of Education。
文摘As a generalization of the two-term conjugate gradient method(CGM),the spectral CGM is one of the effective methods for solving unconstrained optimization.In this paper,we enhance the JJSL conjugate parameter,initially proposed by Jiang et al.(Computational and Applied Mathematics,2021,40:174),through the utilization of a convex combination technique.And this improvement allows for an adaptive search direction by integrating a newly constructed spectral gradient-type restart strategy.Then,we develop a new spectral CGM by employing an inexact line search to determine the step size.With the application of the weak Wolfe line search,we establish the sufficient descent property of the proposed search direction.Moreover,under general assumptions,including the employment of the strong Wolfe line search for step size calculation,we demonstrate the global convergence of our new algorithm.Finally,the given unconstrained optimization test results show that the new algorithm is effective.
文摘Recently, Gilbert and Nocedal([3]) investigated global convergence of conjugate gradient methods related to Polak-Ribiere formular, they restricted beta(k) to non-negative value. [5] discussed the same problem as that in [3] and relaxed beta(k) to be negative with the objective function being convex. This paper allows beta(k) to be selected in a wider range than [5]. Especially, the global convergence of the corresponding algorithm without sufficient decrease condition is proved.
文摘In this paper, we present a new hybrid conjugate gradient algorithm for unconstrained optimization. This method is a convex combination of Liu-Storey conjugate gradient method and Fletcher-Reeves conjugate gradient method. We also prove that the search direction of any hybrid conjugate gradient method, which is a convex combination of two conjugate gradient methods, satisfies the famous D-L conjugacy condition and in the same time accords with the Newton direction with the suitable condition. Furthermore, this property doesn't depend on any line search. Next, we also prove that, moduling the value of the parameter t,the Newton direction condition is equivalent to Dai-Liao conjugacy condition.The strong Wolfe line search conditions are used.The global convergence of this new method is proved.Numerical comparisons show that the present hybrid conjugate gradient algorithm is the efficient one.
基金partially supported by the National Natural Science Foundation of China (No.41230318)
文摘With the development of computational power, there has been an increased focus on data-fitting related seismic inversion techniques for high fidelity seismic velocity model and image, such as full-waveform inversion and least squares migration. However, though more advanced than conventional methods, these data fitting methods can be very expensive in terms of computational cost. Recently, various techniques to optimize these data-fitting seismic inversion problems have been implemented to cater for the industrial need for much improved efficiency. In this study, we propose a general stochastic conjugate gradient method for these data-fitting related inverse problems. We first prescribe the basic theory of our method and then give synthetic examples. Our numerical experiments illustrate the potential of this method for large-size seismic inversion application.
基金Project supported by the National Natural Science Foundation of China(Nos.5130926141030747+3 种基金41102181and 51121005)the National Basic Research Program of China(973 Program)(No.2011CB013503)the Young Teachers’ Initial Funding Scheme of Sun Yat-sen University(No.39000-1188140)
文摘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.
基金Supported by Research Council of Semnan University
文摘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.
基金supported by National Natural Science Foundation of China (Grant No. 51075187)
文摘In one step inverse finite element approach, an initial blank shape is normally predicted from the final deformed shape. The final deformed shape needs to be trimmed into a final part after stamping, the trimmed area, therefore, needs to be compensated manually before using one step inverse approach, which causes low efficiency and in consistency with the real situation. To solve this problem, one step positive approach is proposed to simulate the sheet metal stamping process. Firstly the spatial initial solution of one step positive method is preliminarily obtained by using the mapping relationship and area coordinates, then based on the deformation theory the iterative solving is carried out in three-dimensional coordinate system by using quasi-conjugate-gradient method. During iterative process the contact judgment method is introduced to ensure that the nodes on the spatial initial solution are not separated from die surface. The predicted results of sheet metal forming process that include the shape and thickness of the stamped part can be obtained after the iterative solving process. The validity of the proposed approach is verified by comparing the predicted results obtained through the proposed approach with those obtained through the module of one step inverse approach in Autoform and the real stamped part. In one step positive method, the stamped shape of regular sheet can be calculated fast and effectively. During the iterative solution, the quasi-conjugate-gradient method is proposed to take the place of solving system of equations, and it can improve the stability and precision of the algorithm.
基金Sponsored by the National Natural Science Foundation of China(Grant No.11901561).
文摘Many methods have been put forward to solve unconstrained optimization problems,among which conjugate gradient method(CG)is very important.With the increasing emergence of large⁃scale problems,the subspace technology has become particularly important and widely used in the field of optimization.In this study,a new CG method was put forward,which combined subspace technology and a cubic regularization model.Besides,a special scaled norm in a cubic regularization model was analyzed.Under certain conditions,some significant characteristics of the search direction were given and the convergence of the algorithm was built.Numerical 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.
文摘Conjugate gradient optimization algorithms depend on the search directions with different choices for the parameters in the search directions. In this note, by combining the nice numerical performance of PR and HS methods with the global convergence property of the class of conjugate gradient methods presented by HU and STOREY(1991), a class of new restarting conjugate gradient methods is presented. Global convergences of the new method with two kinds of common line searches, are proved. Firstly, it is shown that, using reverse modulus of continuity function and forcing function, the new method for solving unconstrained optimization can work for a continously dif ferentiable function with Curry-Altman's step size rule and a bounded level set. Secondly, by using comparing technique, some general convergence properties of the new method with other kind of step size rule are established. Numerical experiments show that the new method is efficient by comparing with FR conjugate gradient method.
基金Supported by the Fund of Chongqing Education Committee(KJ091104)
文摘In this paper,an efficient conjugate gradient method is given to solve the general unconstrained optimization problems,which can guarantee the sufficient descent property and the global convergence with the strong Wolfe line search conditions.Numerical results show that the new method is efficient and stationary by comparing with PRP＋ method,so it can be widely used in scientific computation.
基金Supported by the National Natural Science Foundation of China（10571106） Supported by the Fundamental Research Funds for the Central Universities（10CX04044A）
文摘In this note,by combining the nice numerical performance of PR and HS methods with the global convergence property of FR method,a class of new restarting three terms conjugate gradient methods is presented.Global convergence properties of the new method with two kinds of common line searches are proved.
文摘Nonlinear conjugate gradient methods have played an important role in solving large scale unconstrained optimi-zation problems,it is characterized by the simplicity of their iteration and their low memory requirements.It is well-known that the direction generated by a conjugate gradient method may be not a descent direction.In this paper,a new class of nonlinear conjugate gradient method is presented,its search direction is a descent direction for the objective function.If the objective function is differentiable and its gradient is Lipschitz continuous,the line sbarch satisfies strong Wolfe condition,the global convergence result is established.
文摘In this paper, a class of the stochastic generalized linear complementarity problems with finitely many elements is proposed for the first time. Based on the Fischer-Burmeister function, a new conjugate gradient projection method is given for solving the stochastic generalized linear complementarity problems. The global convergence of the conjugate gradient projection method is proved and the related numerical results are also reported.
基金This work is supported by the National Natural Science Foundation of China
文摘In this paper we consider the global convergence of any conjugate gradient method of the form d1=-g1,dk+1=-gk+1+βkdk(k≥1)with any βk satisfying sume conditions,and with the strong wolfe line search conditions.Under the convex assumption on the objective function,we preve the descenf property and the global convergence of this method.
文摘In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method (PCCG). The theorems discuss respectively the qualitative property of the iterative solution and the construction principle of the iterative matrix. The authors put forward a new incompletely LU factorizing technique for non-M-matrix and the method of constructing the iterative matrix. This improved PCCG is used to calculate the ill-conditioned problems and large-scale three-dimensional finite element problems, and simultaneously contrasted with other methods. The abnormal phenomenon is analyzed when PCCG is used to solve the system of ill-conditioned equations, ft is shown that the method proposed in this paper is quite effective in solving the system of large-scale finite element equations and the system of ill-conditioned equations.
文摘In this paper, we propose a method to solve coupled problem. Our computational method is mainly based on conjugate gradient algorithm. We use finite difference method for the structure and finite element method for the fluid. Conjugate gradient method gives suitable numerical results according to some papers.
