Liu Storey Conjugate Gradient Method Research Articles

Exploring Enhanced Conjugate Gradient Methods: A Novel Family of Techniques for Efficient Unconstrained Minimization

Given that the conjugate gradient method (CGM) is computationally efficient and user-friendly, it is often used to address large-scale, unconstrained minimization issues. Numerous researchers have created new conjugate gradient (CG) update parameters by modifying the initial set, also referred to as classical CGMs. This has resulted in the development of several hybrid approaches. This work's major goal is to create a new family of techniques that can be used to create even more new methods. Consequently, Hestenes-Stiefel's update parameter and a new family involving Polak-Ribiere-Polyak and Liu-Storey CGMs are considered. By changing the parameters of this CGM family, a novel approach that possesses sufficient descent characteristics is obtained. A numerical experiment including many unconstrained minimization problems (UMP) is carried out to assess the novel method's efficacy compared to existing approaches. The result reveals that the new CG approach performs better than the current ones.

A three-term conjugate gradient algorithm with restart procedure to solve image restoration problems

Based on the Liu–Storey conjugate gradient method, a three-term conjugate gradient method with restart procedure is proposed for unconstrained optimization. The designed restart condition is chosen according to the Liu–Storey conjugate parameter, and the restart direction consists of the first and third items in the non-restart direction, which can be picked flexibly. Without depending on any line search, the resulting direction satisfies the sufficient descent condition. Moreover, the proposed method possesses the global convergence under usual assumptions and the weak Wolfe line search. Finally, medium–large-scale numerical experiments are conducted for solving unconstrained optimization and image restoration problems, and the numerical results show that the proposed method is promising and practical, even compared with the state-of-the-art methods.

A Liu-Storey-type conjugate gradient method for unconstrained minimization problem with application in motion control

Conjugate gradient methods have played a vital role in finding the minimizers of large-scale unconstrained optimization problems due to the simplicity of their iteration, convergence properties and their low memory requirements. Based on the Liu-Storey conjugate gradient method, in this paper, we present a Liu-Storey type method for finding the minimizers of large-scale unconstrained optimization problems. The direction of the proposed method is constructed in such a way that the sufficient descent condition is satisfied. Furthermore, we establish the global convergence result of the method under the standard Wolfe and Armijo-like line searches. Numerical findings indicate that our presented approach is efficient and robust in solving large-scale test problems. In addition, an application of the method is explored.

A Derivative-Free Liu–Storey Method for Solving Large-Scale Nonlinear Systems of Equations

In this paper, a descent Liu–Storey conjugate gradient method is extended to solve large-scale nonlinear systems of equations. Based on certain assumptions, the global convergence property is obtained with a nonmonotone line search. The proposed method is suitable to solve large-scale problems for the low-storage requirement. Numerical experiment results show that the new method is practically effective.

Modified Three-Term Liu–Storey Conjugate Gradient Method for Solving Unconstrained Optimization Problems and Image Restoration Problems

A new three-term conjugate gradient method is proposed in this article. The new method was able to solve unconstrained optimization problems, image restoration problems, and compressed sensing problems. The method is the convex combination of the steepest descent method and the classical LS method. Without any linear search, the new method has sufficient descent property and trust region property. Unlike previous methods, the information for the function f x is assigned to d k . Next, we make some reasonable assumptions and establish the global convergence of this method under the condition of using the modified Armijo line search. The results of subsequent numerical experiments prove that the new algorithm is more competitive than other algorithms and has a good application prospect.

New Hybrid Conjugate Gradient Method As A Convex Combination of Ls and Fr Methods

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.

Partially symmetrical derivative-free Liu–Storey projection method for convex constrained equations

ABSTRACTApplying Powell symmetrical technique to the Liu–Storey conjugate gradient method, a partially symmetrical Liu–Storey conjugate gradient method is proposed and extended to solve nonlinear monotone equations with convex constraints, which satisfies the sufficient descent condition without any line search. By using some line searches, the global convergence is proved merely by assuming that the equations are Lipschitz continuous. Moreover, we prove the R-linear convergence rate of the proposed method with an additional assumption. Finally, compared with one existing method, the performance of the proposed method is showed by some numerical experiments on the given test problems.

A sufficient descent Liu–Storey conjugate gradient method and its global convergence

In this paper, a modified Liu–Storey conjugate gradient method is proposed. The method can generate sufficient descent directions for non-linear unconstrained optimization problems. A global convergence result is established when the line search fulfils the strong Wolfe conditions. Moreover, the -linear convergence rate of the methods is proved. The extensive numerical results show that the proposed method is efficient for the unconstrained optimization problems in the CUTEr library.

An Liu-Storey-Type Method for Solving Large-Scale Nonlinear Monotone Equations

An algorithm for solving nonlinear monotone equations is proposed, which combines a modified Liu-Storey conjugate gradient method with hyperplane projection method. Under mild conditions, the global convergence of the proposed method is established with a suitable line search method. The method can be applied to solve large-scale problems for its lower storage requirement. Numerical results indicate that our method is efficient.

A sufficient descent LS conjugate gradient method for unconstrained optimization problems

Abstract In this paper, we make a modification to the Liu–Storey (LS) conjugate gradient method and propose a descent LS method. The method can generate sufficient descent directions for the objective function. This property is independent of the line search used. We prove that the modified LS method is globally convergent with the strong Wolfe line search. The numerical results show that the proposed descent LS method is efficient for the unconstrained problems in the CUTEr library.

The convergence of conjugate gradient method with nonmonotone line search

The conjugate gradient method is a useful and powerful approach for solving large-scale minimization problems. Liu and Storey developed a conjugate gradient method, which has good numerical performance but no global convergence under traditional line searches such as Armijo line search, Wolfe line search, and Goldstein line search. In this paper we propose a new nonmonotone line search for Liu-Storey conjugate gradient method (LS in short). The new nonmonotone line search can guarantee the global convergence of LS method and has a good numerical performance. By estimating the Lipschitz constant of the derivative of objective functions in the new nonmonotone line search, we can find an adequate step size and substantially decrease the number of functional evaluations at each iteration. Numerical results show that the new approach is effective in practical computation.

Convergence of Liu–Storey conjugate gradient method

The conjugate gradient method is a useful and powerful approach for solving large-scale minimization problems. Liu and Storey developed a conjugate gradient method, which has good numerical performance but no global convergence result under traditional line searches such as Armijo, Wolfe and Goldstein line searches. In this paper a convergent version of Liu–Storey conjugate gradient method (LS in short) is proposed for minimizing functions that have Lipschitz continuous partial derivatives. By estimating the Lipschitz constant of the derivative of objective functions, we can find an adequate step size at each iteration so as to guarantee the global convergence and improve the efficiency of LS method in practical computation.

A new version of the Liu–Storey conjugate gradient method

In this paper, the global convergence of a new version of the Liu–Storey conjugate gradient method is discussed. This method combines the Liu–Storey conjugate gradient formula and a new inexact line search. We prove that the new method is globally convergent. Some preliminary numerical results show that the corresponding algorithm is efficient.