Fletcher-Reeves Conjugate Gradient Method Research Articles

A modified Fletcher-Reeves conjugate gradient method for unconstrained optimization with applications in image restoration

A modified Fletcher-Reeves conjugate gradient method for unconstrained optimization with applications in image restoration

A modified type of Fletcher-Reeves conjugate gradient method with its global convergence

<p><span>The conjugate gradient methods are one of the most important techniques used to address problems involving minimization or maximization, especially nonlinear optimization problems with no constraints at all. That is because of their simplicity and low memory needed. They can be applied in many areas, such as economics, engineering, neural networks, image restoration, machine learning, and deep learning. The convergence of Fletcher-Reeves (FR) conjugate gradient method has been established under both exact and strong Wolfe line searches. However, it is performance in practice is poor. In this paper, to get good numerical performance from the FR method, a little modification is done. The global convergence of the modified version has been established for general nonlinear functions. Preliminary numerical results show that the modified method is very efficient in terms of number of iterations and CPU time.</span></p>

A class of CG algorithms overcoming jamming of the iterative solving process and its application in image restoration

Intending to improve the numerical performance of the classic Fletcher–Reeves conjugate gradient method, Jiang and Jian embedded a shrinkage multiplier on the Fletcher–Reeves parameter which enhances its computational merits. As a matter of fact, we show that when the iterations jam, the line search turns to be approximately exact and so, search direction of the Jiang–Jian method automatically approaches the steepest descent direction which may lead to overcoming undesirable jamming effect of the classic Fletcher–Reeves method. A more detailed analysis of the numerical behavior of the Fletcher–Reeves method under the jamming phenomenon is presented as well. Then, on account of the advantages of the Jiang–Jian method, a conjugate gradient parameter is devised. Additionally, a descent spectral version of the proposed method is developed which fulfills the sufficient descent property regardless of the line search technique. Convergence of the proposed methods is addressed under classical suppositions. To gain support for our theoretical arguments, the computational merits of the given algorithms on a set of CUTEr test functions and the image restoration problem are depicted.

Development of a New Numerical Conjugate Gradient Technique for Image Processing

We present a new iterative conjugate gradient technique for image processing. The technique is based on a new derivation of the conjugacy coefficient and develops a variant of the classical Fletcher-Reeves conjugate gradient method. The derivation exploits a quadratic function model. The new method is intended to minimize the presence of noise by utilizing the adaptive median filter (AMF) to reduce salt-and-pepper noise, while the adaptive center-weighted median filter (ACWMF) is used to reduce random-valued noise. The theoretical convergence properties of the method are proven and then tested on a basic set of images using MATLAB. The results show that the proposed algorithm is more efficient than the classical Fletcher-Reeves (FR) method, as measured by the signal-to-noise ratio (PSNR). The number of iterations and the number of function evaluations are also lower for the proposed method. The favorable performance of the new algorithm provides promise for deriving similar techniques that enhance the speed and efficiency of image-processing libraries.

A Nonmonotone Scaled Fletcher–Reeves Conjugate Gradient Method with Application in Image Reconstruction

A Nonmonotone Scaled Fletcher–Reeves Conjugate Gradient Method with Application in Image Reconstruction

Assessment and mitigation of DC bias risk in power grids at provincial boundaries

With the wide application of high voltage/ultra-high voltage (HV/UHV) DC transmission technology, the impact of DC grounding electrode location selection on the surrounding power grid has become increasingly prominent, especially the problem of DC bias hazard caused by DC grounding electrodes at provincial grid boundaries needs to be solved urgently. This paper studies the assessment and prevention of trans-regional DC bias risk, and proposes an inversion method of earth resistivity model based on the measured data of neutral current in the provincial boundary area. Firstly, the DC bias risk of provincial boundary power grid is simulated and calculated, and the influence of reasonable selection of earth model on the accuracy of risk assessment results is explained. Based on the classical Fletcher-Reeves conjugate gradient method and the measured neutral current data, a reduced-order inversion method for the earth resistivity parameters of the provincial boundary power grid is proposed. On this basis, a set of DC bias risk control scheme is formulated for the actual project of Gannan power grid. Finally, the feasibility of the scheme is verified by simulation and measured data.

A Modified Fletcher–Reeves Conjugate Gradient Method for Monotone Nonlinear Equations with Some Applications

One of the fastest growing and efficient methods for solving the unconstrained minimization problem is the conjugate gradient method (CG). Recently, considerable efforts have been made to extend the CG method for solving monotone nonlinear equations. In this research article, we present a modification of the Fletcher–Reeves (FR) conjugate gradient projection method for constrained monotone nonlinear equations. The method possesses sufficient descent property and its global convergence was proved using some appropriate assumptions. Two sets of numerical experiments were carried out to show the good performance of the proposed method compared with some existing ones. The first experiment was for solving monotone constrained nonlinear equations using some benchmark test problem while the second experiment was applying the method in signal and image recovery problems arising from compressive sensing.

A Nonlinear Conjugate Gradient Algorithm under Strong Wolfe-Powell Line Search for Large Scale Unconstrained Optimization Problems

The conventional conjugate gradient method solves linear and quadratic optimization problems but most real life problems consist of nonquadratic functions of several variables. In this work a nonlinear conjugate gradient algorithm for solving large scale optimization problems is presented. The new algorithm is a modification of the Fletcher-Reeves conjugate gradient method and it is proved to achieve global convergence under the strong Wolfe-Powell inexact line search technique. Computational experiments show that the new algorithm presented performs better than the Fletcher-Reeves exact line search algorithm in solving high dimensional nonlinear optimization problems.

Global convergence of a modified Fletcher–Reeves conjugate gradient method with Wolfe line search

In this paper, we are concerned with the conjugate gradient methods for solving unconstrained optimization problems. we propose a modified Fletcher–Reeves (abbreviated FR) [Function minimization by conjugate gradients, Comput. J. 7 (1964) 149–154] conjugate gradient algorithm satisfying a parametrized sufficient descent condition with a parameter [Formula: see text] is proposed. The parameter [Formula: see text] is computed by means of the conjugacy condition, thus an algorithm which is a positive multiplicative modification of the Hestenes and Stiefel (abbreviated HS) [Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards Sec. B 48 (1952) 409–436] algorithm is obtained, which produces a descent search direction at every iteration that the line search satisfies the Wolfe conditions. Under appropriate conditions, we show that the modified FR method with the strong Wolfe line search is globally convergent of uniformly convex functions. We also present extensive preliminary numerical experiments to show the efficiency of the proposed method.

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.

Evaluation on Partial Discharge Intensity of Electrical Equipment Based on Improved ANFIS and Ultraviolet Pulse Detection Technology

The ultraviolet pulse detection technology has superiorities in response speed and antijamming capability, so it is extensively applied to insulation detection for electrical equipment. The existing detection and analysis methods have defects, such as failure of precise modeling caused by incomplete analysis of influence factor, poor model adaptability, and complex operation. In allusion to the problems, the ultraviolet pulse detection circuit was optimized and its sensitivity was analyzed through tests. Then, a partial discharge intensity evaluation method for electrical equipment based on the improved adaptive network based fuzzy inference system (improved ANFIS) was proposed combined with the detected pulse count (P), temperature (T) and humidity (H). The initial fuzzy inference system structure was established with the subtractive clustering method (SCM) and fuzzy C-means (FCM) algorithm, and the traditional ANFIS learning algorithm was improved via Fletcher-Reeves conjugate gradient method. In this way, the model parameters were optimized continuously, and the system ability of ignoring small changes in the network was improved. Finally, the effectiveness and practicability of the method were verified through field test. The experimental results demonstrated that the improved ANFIS reduced the model error by 2% when compared with traditional ANFIS, and the model accuracy is improved. Besides, the quantitative precision of the discharge intensity is higher than that of the traditional ANFIS by the contrastive analysis of field test data, indicating that the improved ANFIS evaluate the partial discharge intensity of electrical equipment more accurately. Thus, a decision basis can be provided for the equipment protection and charged maintenance of the electric system.

A Riemannian variant of the Fletcher–Reeves conjugate gradient method for stochastic inverse eigenvalue problems with partial eigendata

SummaryIn this paper, we focus on the stochastic inverse eigenvalue problem with partial eigendata of constructing a stochastic matrix from the prescribed partial eigendata. A Riemannian variant of the Fletcher–Reeves conjugate gradient method is proposed for solving a general unconstrained minimization problem on a Riemannian manifold, and the corresponding global convergence is established under some assumptions. Then, we reformulate the inverse problem as a nonlinear least squares problem over a matrix oblique manifold, and the application of the proposed geometric method to the nonlinear least squares problem is investigated. The proposed geometric method is also applied to the case of prescribed entries and the case of column stochastic matrix. Finally, some numerical tests are reported to illustrate that the proposed geometric method is effective for solving the inverse problem.

Memetic reconstruction algorithm for the ECT

The practical performance of the electrical capacitance tomography (ECT) instrument largely relies on the accuracy and robustness of the image reconstruction algorithm. Varying from the common imaging methods, in this study, a regularisation-based loss function is proposed to transform the ECT imaging task into an optimisation problem, in which the least trimmed squares method is employed to measure the data fidelity, and the l p -norm of the variables is used as the regulariser. An efficient memetic algorithm (MA) that constitutes of the monkey king evolution algorithm and the Fletcher-Reeves conjugate gradient method is developed for solving the proposed loss function. The numerical simulations reveal that the feasibility and robustness of the proposed MA lead to a series of satisfactory results in the ECT field.

A modified quadratic hybridization of Polak-Ribiere-Polyak and Fletcher-Reeves conjugate gradient method for unconstrained optimization problems

This article presents a modified quadratic hybridization of the Polak–Ribiere–Polyak and Fletcher–Reeves conjugate gradient method for solving unconstrained optimization problems. Global convergence, with the strong Wolfe line search conditions, of the proposed quadratic hybrid conjugate gradient method is established. We also report some numerical results to show the competitiveness of the new hybrid method.

A class of modified FR conjugate gradient method and applications to non-negative matrix factorization

Conjugate gradient method, as an efficient method, is used to solve unconstrained optimization problems. In this paper, we propose a class of modified Fletcher–Reeves conjugate gradient method, with Armijo-type line search, which generates the direction is descent for the objective function. Under mild conditions, we give that the proposed method is descent and globally convergent. Moreover, the method is applied to nonnegative matrix factorization, and the experimental results demonstrate the validity of the proposed method.

A MODIFIED FR CONJUGATE GRADIENT METHOD FOR COMPUTING -EIGENPAIRS OF SYMMETRIC TENSORS

This paper proposes improvements to the modified Fletcher–Reeves conjugate gradient method (FR-CGM) for computing $Z$-eigenpairs of symmetric tensors. The FR-CGM does not need to compute the exact gradient and Jacobian. The global convergence of this method is established. We also test other conjugate gradient methods such as the modified Polak–Ribière–Polyak conjugate gradient method (PRP-CGM) and shifted power method (SS-HOPM). Numerical experiments of FR-CGM, PRP-CGM and SS-HOPM show the efficiency of the proposed method for finding $Z$-eigenpairs of symmetric tensors.

Integrated Mathematical Model of the Processes in the Turbine with Adjustable Steam Extraction

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An extension of the Fletcher–Reeves method to linear equality constrained optimization problem

In this paper, combining the feasible direction method with the conjugate gradient method, we propose a feasible Fletcher–Reeves conjugate gradient method for solving linear equality constrained optimization problem. The directions generated by the method are feasible and descent for the objective function. Under mild conditions, we show that the proposed method with exact line search is globally convergent. Numerical experiments are also given to show the efficiency of the method.

New step lengths in conjugate gradient methods

Consider any conjugate gradient method for finding a zero point of a given gradient whose function is implicit. We propose two different types of conditions for selecting the step length using the gradient information only. One is used for re-proving known convergence results under the same gradient-Lipschitz assumption. Moreover, if the gradient is merely continuous then we are still able to get some interesting convergence results. The other also allows for convergence of the resulting conjugate gradient methods, with an application to convergence analysis of the Fletcher–Reeves conjugate gradient method. Preliminary numerical experiments show the efficiency of our proposed step length rules in practice.

An inverse analysis for the estimation of boundary heat flux in a circular pipe employing the proper orthogonal decomposition

An inverse algorithm based on the proper orthogonal decomposition technique is developed in this paper. The performance of the present inverse algorithm is examined by an inverse forced convection problem of identifying the unknown space-dependent heat flux at the outer boundary of a circular pipe. The inverse problem is resolved in a functional optimisation way by the Fletcher-Reeves conjugate gradient method. The effects of the convection, the location of the thermocouples and the measurement error on the performance of the inverse algorithm are studied thoroughly. It was shown that the present POD-based inverse algorithm is very accurate as well as efficient. A large time saving is also found by using the present inverse algorithm.