Conjugate Gradient Coefficient Research Articles

A New Hybrid Three-Term LS-CD Conjugate Gradient In Solving Unconstrained Optimization Problems

The Conjugate Gradient (CG) method is renowned for its rapid convergence in optimization applications. Over the years, several modifications to CG methods have emerged to improve computational efficiency and tackle practical challenges. This paper presents a new three-term hybrid CG method for solving unconstrained optimization problems. This algorithm utilizes a search direction that combines Liu-Storey (LS) and Conjugate Descent (CD) CG coefficients and standardizes it using a spectral which acts as a scheme for the choices of the conjugate parameters. This resultant direction closely approximates the memoryless Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton direction, known for its bounded nature and compliance with the sufficient descent condition. The paper establishes the global convergence under standard Wolfe conditions and some appropriate assumptions. Additionally, the numerical experiments were conducted to emphasize the robustness and superior efficiency of this hybrid algorithm in comparison to existing approaches.

Optimization of unconstrained problems using a developed algorithm of spectral conjugate gradient method calculation

This paper presents a numerical investigation of the spectral conjugate directions formulation for optimizing unconstrained problems. A novel modified algorithm is proposed based on the conjugate gradient coefficient method. The algorithm employs the Wolfe inexact line search conditions to determine the optimum step length at each iteration and selects the appropriate conjugate gradient coefficient accordingly. The algorithm is evaluated through several numerical experiments using various unconstrained functions. The results indicate that the algorithm is highly stable, regardless of the starting point, and has better convergence rates and efficiency compared to classical methods in certain cases. Overall, this research provides a promising approach to solving unconstrained optimization problems.

A new family of hybrid three-term conjugate gradient method for unconstrained optimization with application to image restoration and portfolio selection

<abstract><p>The conjugate gradient (CG) method is an optimization method, which, in its application, has a fast convergence. Until now, many CG methods have been developed to improve computational performance and have been applied to real-world problems. In this paper, a new hybrid three-term CG method is proposed for solving unconstrained optimization problems. The search direction is a three-term hybrid form of the Hestenes-Stiefel (HS) and the Polak-Ribiére-Polyak (PRP) CG coefficients, and it satisfies the sufficient descent condition. In addition, the global convergence properties of the proposed method will also be proved under the weak Wolfe line search. By using several test functions, numerical results show that the proposed method is most efficient compared to some of the existing methods. In addition, the proposed method is used in practical application problems for image restoration and portfolio selection.</p></abstract>

A Three-Term Conjugate Gradient Method for Non-Convex Functions with Applications for Heat Transfer

The problem of unconstrained optimization (UOP) has recently gained a great deal of attention from researchers around the globe due to its numerous real-life applications. The conjugate gradient (CG) method is among the most widely used algorithms for solving UOP because of its good convergence properties and low memory requirements. This study investigates the performance of a modified CG coefficient for optimization functions, proof of sufficient descent, and global convergence of the new CG method under suitable, standard Wolfe conditions. Computational results on several benchmark problems are presented to validate the robustness and efficacy of the new algorithm. The proposed method was also applied to solve function estimations in inverse heat transfer problems. Another interesting feature possessed by the proposed modification is the ability to solve problems on a large scale and use different dimensions. Based on the theoretical and computational efficiency of the new method, we can conclude that the new coefficient can be a better alternative for solving unconstrained optimization and real-life application problems. KEYWORDS Computational efficiency, global convergence, inverse heat, low memory, optimization problems, theoretical efficiency

Modified spectral conjugate gradient iterative scheme for unconstrained optimization problems with application on COVID-19 model

In this work, a new class of spectral conjugate gradient (CG) method is proposed for solving unconstrained optimization models. The search direction of the new method uses the ZPRP and JYJLL CG coefficients. The search direction satisfies the descent condition independent of the line search. The global convergence properties of the proposed method under the strong Wolfe line search are proved with some certain assumptions. Based on some test functions, numerical experiments are presented to show the proposed method's efficiency compared with other existing methods. The application of the proposed method for solving regression models of COVID-19 is provided.Mathematics subject classification65K10, 90C52, 90C26.

A new modified conjugate gradient method under the strong Wolfe line search for solving unconstrained optimization problems

Conjugate gradient (CG) method is well-known due to efficiency to solve the problems of unconstrained optimization because of its convergence properties and low computation cost. Nowadays, the method is widely developed to compete with existing methods in term of their efficiency. In this paper, a modification of CG method will be proposed under strong Wolfe line search. A new CG coefficient is presented based on the idea of make use some parts of the previous existing CG methods to retain the advantages. The proposed method guarantees that the sufficient descent condition holds and globally convergent under inexact line search. Numerical testing provides strong indication that the proposed method has better capability when solving unconstrained optimization compared to the other methods under inexact line search specifically strong Wolfe–Powell line search.

A new class of nonlinear conjugate gradient coefficients for unconstrained optimization

The nonlinear Conjugate gradient method (CGM) is a very effective way in solving large-scale optimization problems. Zhang et al. proposed a new CG coefficient which is defined by [Formula: see text]. They proved the sufficient descent condition and the global convergence for nonconvex minimization in strong Wolfe line search. In this paper, we prove that this CG coefficient possesses sufficient descent conditions and global convergence properties under the exact line search.

Performance Analysis of a Modified Conjugate Gradient Algorithm for Optimization Models

The Conjugate gradient (CG) algorithms is very important and widely used in solving optimization models. This is due to its simplicity as well as global convergence properties. Various line search procedures as usually employ in the analysis of the CG methods. Recently, many studies have been done aimed at improving the CG method. In this paper, an alternative formula for conjugate gradient coefficient has been proposed which possesses the global convergence properties under exact minimization condition. The result of the numerical computation has shown that this new coefficient performs better than the existing CG methods.

A COMPARATIVE STUDY OF SOME MODIFICATIONS OF CG METHODS UNDER EXACT LINE SEARCH

Conjugate Gradient (CG) method is a technique used in solving nonlinear unconstrained optimization problems. In this paper, we analysed the performance of two modifications and compared the results with the classical conjugate gradient methods of. These proposed methods possesse global convergence properties for general functions using exact line search. Numerical experiments show that the two modifications are more efficient for the test problems compared to classical CG coefficients.

An Alternative Modified Conjugate Gradient Coefficient for Solving Nonlinear System of Equations

In mathematical term, the method of solving models and finding the best alternatives is known as optimization. Conjugate gradient (CG) method is an evolution of computational method in solving optimization problems. In this article, an alternative modified conjugate gradient coefficient for solving large-scale nonlinear system of equations is presented. The method is an improved version of the Rivaie et el conjugate gradient method for unconstrained optimization problems. The new CG is tested on a set of test functions under exact line search. The approach is easy to implement due to its derivative-free nature and has been proven to be effective in solving real-life application. Under some mild assumptions, the global convergence of the proposed method is established. The new CG coefficient also retains the sufficient descent condition. The performance of the new method is compared to the well-known previous PRP CG methods based on number of iterations and CPU time. Numerical results using some benchmark problems show that the proposed method is promising and has the best efficiency amongst all the methods tested.

New class of hybrid conjugate gradient coefficients with guaranteed descent and efficient line search

Hybrid conjugate gradient (CG) techniques are one of the most prominent procedure for obtaining the solution of large-scale unconstrained optimization problems. This is due to its simplicity, global convergence, and low memory requirement. Numerous modifications have been done recently to improve the performance of these methods. In this paper, we proposed new class of hybrid CG coefficients with guaranteed descent under exact line search. Numerical results are presented to illustrate the efficiency of the proposed methodscompared to other classical CG coefficients.

A New Search Direction via Hybrid Conjugate Gradient Coefficient for Solving Nonlinear System of Equations

Conjugate gradient methods are widely used for unconstrained optimization problems. Most of the conjugate gradient methods do not always generate a descent search direction. In this article, a new search direction via hybrid conjugate gradient method by convex combination of the earlier works is used and extended for the solution of nonlinear system of equations. The proposed search direction generates a descent direction at every iteration and without convexity assumption on the objective function. The new method possesses, preserved and inherits the properties of the Conjugate Gradient (CG) coefficient used. The proposed hybridized method played a useful, vital and powerful role in solving large-scale nonlinear system of equations as indicated in the numerical results.

A New Conjugate Gradient Method with Exact Line Search

The nonlinear conjugate gradient (CG) method is a widely used approach for solving large-scale optimization problems in many fields, such as physics, engineering, economics, and design. The efficiency of this method is mainly attributable to its global convergence properties and low memory requirement. In this paper, a new conjugate gradient coefficient is proposed based on the Aini-Rivaie-Mustafa (ARM) method. Furthermore, the proposed method is proved globally convergent under exact line search. This is supported by the results of the numerical tests. The numerical performance of the new CG method better than other related and more efficient compared with previous CG methods.

The Sufficient Descent Condition of Nonlinear Conjugate Gradient Method

Non-linear conjugate gradient methods has been widely used instrumental in solving large scale optimization. These methods has been proved that only required very low memory other than its numerical efficiency. Thus, many studies have been conducted to improve these methods to find the most efficient method. In this paper, we proposed a new non-linear conjugate gradient coefficient that guarantees sufficient descent condition. Numerical tests indicate that the proposed coefficient is better than the three classical conjugate gradient coefficients.

Approximating the extreme Ritz values and upper bounds for the A-norm of the error in CG

In practical conjugate gradient (CG) computations, it is important to monitor the quality of the approximate solution to Ax = b so that the CG algorithm can be stopped when the required accuracy is reached. The relevant convergence characteristics, like the A-norm of the error or the normwise backward error, cannot be easily computed. However, they can be estimated. Such estimates often depend on approximations of the smallest or largest eigenvalue of A. In the paper, we introduce a new upper bound for the A-norm of the error, which is closely related to the Gauss-Radau upper bound, and discuss the problem of choosing the parameter µ which should represent a lower bound for the smallest eigenvalue of A. The new bound has several practical advantages, the most important one is that it can be used as an approximation to the A-norm of the error even if µ is not exactly a lower bound for the smallest eigenvalue of A. In this case, µ can be chosen, e.g., as the smallest Ritz value or its approximation. We also describe a very cheap algorithm, based on the incremental norm estimation technique, which allows to estimate the smallest and largest Ritz values during the CG computations. An improvement of the accuracy of these estimates of extreme Ritz values is possible, at the cost of storing the CG coefficients and solving a linear system with a tridiagonal matrix at each CG iteration. Finally, we discuss how to cheaply approximate the normwise backward error. The numerical experiments demonstrate the efficiency of the estimates of the extreme Ritz values, and show their practical use in error estimation in CG.

A modified conjugate gradient coefficient under exact line search for unconstrained optimization

The conjugate gradient (CG) method is a well-known solver for large-scale unconstrained optimization problems. In this paper, a modified CG method based on AMR* and CD method is presented. The resulting algorithm for the new CG method is proved to be globally convergent under exact line search both under some mild conditions. Comparisons of numerical performance are made involving the new method and four other CG methods. The results show that the proposed method is more efficient.

A new family of conjugate gradient coefficient with application

Conjugate gradient (CG) methods are famous for their utilization in solving unconstrained optimization problems, particularly for large scale problems and have become more intriguing such as in engineering field. In this paper, we propose a new family of CG coefficient and apply in regression analysis. The global convergence is established by using exact and inexact line search. Numerical results are presented based on the number of iterations and CPU time. The findings show that our method is more efficient in comparison to some of the previous CG methods for a given standard test problems and successfully solve the real life problem.

New Hybrid Conjugate Gradient Method with Global Convergence Properties for Unconstrained Optimization

Nonlinear conjugate gradient (CG) method holds an important role in solving large-scale unconstrained optimization problems. In this paper, we suggest a new modification of CG coefficient that satisfies sufficient descent condition and possesses global convergence property under strong Wolfe line search. The numerical results show that our new method is more efficient compared with other CG formulas tested.

The application of new conjugate gradient methods in estimating data

Many researchers are intended to improve the conjugate gradient (CG) methods as well as their applications in real life. Besides, CG become more interesting and useful in many disciplines and has important role for solving large-scale optimization problems. In this paper, three types of new CG coefficients are presented with application in estimating data. Numerical experiments show that the proposed methods have succeeded in solving problems under strong Wolfe Powell line search conditions.

A new class of nonlinear conjugate gradient with global convergence properties

Nonlinear conjugate gradient methods are widely used in solving large scale unconstrained optimization. Their wide application in many fields are due to their low memory. Numerous studies have been conducted recently to improve these methods. In this paper, a new class of conjugate gradient coefficients that possess global convergence properties is proposed. The global convergence result using exact line searches are discussed. Numerical result shows that the proposed method is more efficient compared with the other classical conjugate gradient.