In this paper, we propose a new derivativefree preconditioned conjugate gradient method in order for solving largescale square and underdetermined nonlinear systems of equations. By making use of the moreauyosida regularization, a nonmonotone line search technique of and a new secant equation of derived by the authors earlier, we present a modified prp conjugate gradient algorithm for solving nonsmooth convex optimization problems. However, some conjugate gradient methods have no global convergence. Li and yang journal of inequalities and applications a nonmonotone hybrid conjugate gradient method for unconstrained optimization wenyu li 0 yueting yang 0 0 school of mathematics and statistics, beihua university, jilin street no.
Yuan, a nonlinear conjugate gradient with a strong global convergence properties, siam journal of optimization, 10, pp. The conjugate gradient cg method is one of the most popular methods for solving smooth unconstrained optimization problems due to its simplicity and low memory requirement. The source code and files included in this project are listed in the project files section, please make sure whether the listed source code meet your needs there. Zhang, a new conjugate gradient method with guaranteed. The result is conjugate gradient on the normal equations cgnr. A fast conjugate gradient algorithm with active set prediction for. The sigma plotting software was used to graph the data. An introduction to the conjugate gradient method without the agonizing pain edition 11 4 jonathan richard shewchuk august 4, 1994 school of computer science carnegie mellon university pittsburgh, pa 152 abstract the conjugate gradient method is the most prominent iterative method for solving sparse systems of linear equations. A hybrid method combining the fr conjugate gradient method and the wyl conjugate gradient method is proposed for unconstrained optimization problems. A new class of conjugate gradient methods with extended. The new nonmonotone line search needs to estimate the lipschitz constant of the. A nonmonotone prp conjugate gradient method for solving. Unconstrained minimization of a nonlinear smooth function.
In this paper conjugate gradient methods with nonmonotone line search technique are introduced. This article studies the convergence behavior of the algorithm. A modified polakribierepolyak conjugate gradient algorithm. The modified hz conjugate gradient algorithm for large. Instead of using the residual and its conjugate, the cgs algorithm avoids using the transpose of the coefficient matrix by working with a squared residual 1. Acm transactions on mathematical software, 32 2006, pp. Based on the theories of time series analysis and unconstrained optimization, a new spectral conjugate gradient methodautoregressive integrated moving average combined model fhs spectral cgarima combined model is proposed to fit and predict the actual time series. The method incorporates the modified bfgs secant equation in an effort to include the second order information of the objective function. A hybrid conjugate gradient method for optimization problems. Theory of algorithms for unconstrained optimization acta. This problem is avoided in the conjugate gradient cg method, which does not repeat any previous search direction and converge in iterations.
Two new prp conjugate algorithms are proposed in this paper based on two modified prp conjugate gradient methods. Finally, in order to separately assess the nonmonotone strategy and the preconditioned conjugate gradient technique, we compare the standard nonmonotone version of dfuprp algorithm with a spectral residual version of the algorithm which is obtained by setting. The nonlinear conjugate gradient cg method for is designed by the iterative form where is the th iterative point, is a steplength, and is the search direction defined by where is a scalar which determines the different conjugate gradient methods 1, 2, and is the gradient of at the point. Solve system of linear equations conjugate gradients. Dai, a nonmonotone conjugate gradient algorithm for unconstrained optimization, journal of systems science and complexity, 15, pp. A conjugate gradient method for unconstrained optimization. Under some mild conditions, convergent results of the proposed methods are established. A hybrid method combining the fr conjugate gradient method and the wyl conjugate. Conjugate gradient in matlab download free open source.
A modified threeterm prp conjugate gradient algorithm for. A few months ago, while preparing a lecture to an audience that included engineers and numerical analysts, i asked myself the question. Of course, as the software developed in the framework of the mitiv project, icy is public domain and thus freely available. This technique is generally used as an iterative algorithm, however, it can be used as a direct method, and it will produce a numerical solution. A nonmonotone hybrid conjugate gradient method for unconstrained. Hager, a nonmonotone line search technique and its. The conjugate gradient cg method is one of the most popular methods for solving.
The conjugate gradients squared cgs algorithm was developed as an improvement to the biconjugate gradient bicg algorithm. Conjugate gradient matlab code download free open source. The conjugate gradient method can be applied to an arbitrary nbym matrix by applying it to normal equations a t a and righthand side vector a t b, since a t a is a symmetric positivesemidefinite matrix for any a. A hybrid method of the polakribierepolyak prp method and the weiyaoliu wyl method is proposed for unconstrained optimization pro blems, which possesses the following properties. China 2 laboratoire collisions agrgats ractivit, universit paul sabatier, 31062 toulouse cedex 09, france. Abstractthis paper proposes a nonmonotone scaled conjugate gradient algorithm for solving largescale unconstrained optimization problems, which. It is of great practical significance to fit and predict actual time series. This paper presents a nonmonotone scaled memoryless bfgs preconditioned conjugate gradient algorithm for solving nonsmooth convex optimization problems, which combines the idea of scaled memoryless bfgs preconditioned conjugate gradient method with the nonmonotone technique and the moreauyosida regularization. Some numerical experiments indicate that the proposed method is superior to the limited memory conjugate gradient software package cg descent 6. The technique of nonmonotone line search has received many successful applications and extensions in nonlinear optimization. Zhang, a survey of nonlinear conjugate gradient methods, pacific journal of optimization, 2 2006, pp. These regularization techniques are based on the strategy of computing an approximate global minimizer of a cubic overestimator of the objective function. On the convergence of a new conjugate gradient algorithm.
In practice, the cg algorithm is widely used, but it is not suitable when the hessian matrix is indefinite, as it can stop prematurely. In this paper, we provide and analyze a new scaled conjugate gradient method and its performance, based on the modified secant equation of the broydenfletchergoldfarbshanno bfgs method and on a new modified nonmonotone line search technique. Raydan universidad simon bol var abstract over the last two decades, it has been observed that using the gradient vector as a search direction in largescale optimization may lead to e cient algorithms. Global convergence properties of conjugate gradient. In this paper a new nonmonotone conjugate gradient method is introduced, which can be regarded as a generalization of the perry and shanno memoryless quasinewton method. In general, vmlm is more efficient than nlcg but may require more memory. Nonmonotone spectral methods for largescale nonlinear. A new nonmonotone spectral conjugate gradient method for. Our rst proposed algorithm combines the spectral conjugate gradient of birgin and mart nez 8 with an adaptive nonmonotone strategy 29, exploiting the e ciency of the spectral conjugate gradient algorithm and the tuning strategy for the nonmonotone learning horizon. A derivativefree prp method for solving largescale nonlinear. A new nonmonotone adaptive retrospective trust region method for unconstrained optimization problems. Our method satisfies the sufficiently descent property automatically, and the.
The approximate solution must satisfy suitable properties to ensure global convergence. Reconstruction of fluorescence molecular tomography via a. A conjugate gradient type method for the nonnegative constraints optimization problems li, can, journal of applied mathematics, 20. A nonmonotone prp conjugate gradient method for solving square and underdetermined systems of equations. This is a brief introduction to the optimization algorithm called conjugate gradient. A nonmonotone hybrid conjugate gradient method is proposed, in which the technique of the nonmonotone wolfe line search is used. While the second one revises the adaptive nonmonotone self. Read new nonlinear conjugate gradient formulas for largescale unconstrained optimization problems, applied mathematics and computation on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. For convex objective functions, the proposed nonmonotone conjugate gradient method is proved to be globally convergent. The generated search direction satisfies both the sufficient descent condition and the dailiao conjugacy condition independent of line search.
The spectral gradient method has proved to be effective for solving largescale optimization problems. In recent years, cubic regularization algorithms for unconstrained optimization have been defined as alternatives to trustregion and line search schemes. A truncated newton method consists of repeated application of an iterative optimization algorithm to approximately solve newtons equations, to determine an update to the functions parameters. An efficient barzilaiborwein conjugate gradient method. A nonmonotone hybrid conjugate gradient method for.
The cga is only slightly more complicated to implement than the method of steepest descent but converges in a finite number of steps on quadratic problems. Hello, parallel implementation of conjugate gradient linear system solver 1. Global convergence properties of conjugate gradient methods. The theory, derivations to the fast implementation and an interactive example are found here. Furthermore, the presented method has sufficiently descent property and characteristic of. The conjugate gradient method finds the solution of a linear system of equations by stepping to the solution in conjugate directions. Acm transactions on mathematical software, 21, 123160. An extended polakribierepolyak conjugate gradient method for solving nonlinear systems of equations is.
Dai, a nonmonotone conjugate gradient algorithm for unconstrained. The proposed method makes use of approximate function and gradient. Journal of systems science and complexity, 15, 9145. The algorithms developed in the mitiv project are based on inverse approach and require the minimization of large problems e. Gradient descent and conjugate gradient descent stack exchange. The analyses are helpful in establishing the global convergence of a. This derivativefree feature of the proposed method gives it advantage to solve relatively largescale problems 500,000 variables with lower storage requirement compared to some existing methods. Nonmonotone conjugate gradient methods for optimization. Dec 11, 20 a brief overview of steepest descent and how it leads the an optimization technique called the conjugate gradient method. Conjugate gradient the source code and files included in this project are listed in the project files section, please make sure whether the listed source code meet your needs there. Conjugate gradient cgtype method for the solution of. J benchmarking optimization software with performance profiles. A modified polakribierepolyak conjugate gradient algorithm for.
In this work we extend the spectral approach to solve nonlinear systems of equations. Solve system of linear equations preconditioned conjugate. Specifically, we analyze the nonmonotone line search methods for general nonconvex functions along different lines. R e p o r t a survey on algorithms for training artificial. In contrast to newton method, there is no need for matrix inversion. On the proximal gradient algorithm with alternated inertia 2018. There are two methods in optimpack to minimize a nonlinear smooth multivariate function without constraints. An introduction to the conjugate gradient method without the. A cubic regularization algorithm for unconstrained. A nonlinear conjugate gradient algorithm with an optimal property and an improved wolfe line search. Dec 12, 20 this is a brief introduction to the optimization algorithm called conjugate gradient. Icy is written in java and tipi is the core software that we had to develop to implement our algorithms and achieve good performances. Siam journal on numerical analysis society for industrial.
Cg conjugate gradient cg solver for linear systems. A nonmonotone line search method for regression analysis. A modified prp conjugate gradient method, annals of. The following matlab project contains the source code and matlab examples used for conjugate gradient. Benchmarking optimization software with performance. We show that performance profiles combine the best features of other tools for performance evaluation. A new subspace minimization conjugate gradient algorithm with a nonmonotone wolfe line search is proposed and analyzed. An introduction to the conjugate gradient method without the agonizing pain jonathan richard shewchuk march 7, 1994 cmucs94125 school of computer science carnegie mellon university pittsburgh, pa 152 abstract the conjugategradient method is themost prominent iterativemethod for solvingsparse systems of linear equations. The spectral gradient and conjugate gradient methods are a class of methods that can suitably cope with largescale settings. A nonmonotone hybrid conjugate gradient method is proposed, in which the. Also shows a simple matlab example of using conjugate gradient to solve a. A class of accelerated conjugategradientlike methods based. A derivativefree conjugate gradient method and its global. Cg is a fortran90 library which implements a simple version of the conjugate gradient cg method for solving a system of linear equations of the form axb, suitable for situations in which the matrix a is positive definite only real, positive eigenvalues and symmetric.
Conjugate gradient methods are iterative methods for finding the minimizer of a scalar function fx of a vector variable x which do not update an approximation to the inverse hessian matrix. In particular, polakribierepolyak and liustorey conjugate gradient methods are special cases of the new class of conjugate gradient methods. The analyses are helpful in establishing the global convergence of a nonmonotone. A new class of conjugate gradient methods with extended nonmonotone line search hailin liu1 and xiaoyong li2 1 school of computer science,guangdong polytechnic normal university,guangzhou, guangdong 510665, p. Under some suitable assumptions, the global convergence property is established. A scaled conjugate gradient method based on new bfgs secant. Moreover, the value of the parameter contains more useful information without adding more computational cost. Gradient descent is the method that iteratively searches for a minimizer by looking in the gradient direction. This paper provides some basic analyses of the nonmonotone line search. Under mild assumptions, we prove the global convergence and linear convergence rate of the method. Optimpack library claude bernard university lyon 1. In this paper, the hager and zhang hz conjugate gradient cg method and the modified hz mhz cg method are presented for largescale nonsmooth convex minimization.
A new spectral conjugate gradient method and arima. A scaled conjugate gradient method based on new bfgs. In this work we focus on the adaptive regularization algorithm using cubics arc. Pdf a new derivativefree conjugate gradient method for. A new nonmonotone line search technique is proposed to guarantee the global convergence of these conjugate gradient methods under some mild conditions. Abstract pdf 679 kb 2017 a nonmonotone prp conjugate gradient method for solving square and underdetermined systems of equations.
Parallel implementation of conjugate gradient linear system. This class of nonmonotone conjugate gradient methods is proved to be globally convergent when it is. The preconditioned conjugate gradients method pcg was developed to exploit the structure of symmetric positive definite matrices. This new line search technique is based on a relaxation of the strong wolfe conditions and it allows to accept larger steps. We propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software. Rlinear convergence of the barzilai and borwein gradient method.
The conjugate gradient method is a mathematical technique that can be useful for the optimization of both linear and nonlinear systems. This paper proposes a new class of accelerated conjugategradientlike algorithms for solving large scale unconstrained optimization problems, which combine the idea of accelerated adaptive perry conjugate gradient algorithms proposed by andrei 2017 with the modified secant condition and the nonmonotone line search technique. Birgin university of sao paulo jos e mario mart nez university of campinas marcos raydan universidad simon bol var abstract over the last two decades, it has been observed that using the gradient vector as a search direction in largescale optimization may lead to e cient. The spectral gradient method 9 has been successfully extended in 10 for solving square nonlinear systems of equations using grippos nonmonotone line search technique 11. A simulated annealingbased barzilaiborwein gradient. Unlike traditional trust region methods, the subproblem in our method is a simple conic model, where the hessian of the objective function is approximated by a scalar matrix. The limited memory conjugate gradient method request pdf. A class of accelerated conjugategradientlike methods. A class of nonmonotone conjugate gradient methods for. This paper examines the effects of inexact linear searches on the methods and shows how the traditional fletcherreeves and polakribiere algorithm may be modified in a form discovered by perry to a.
Nonmonotone adaptive trust region method based on simple. The nonlinear conjugate gradient cg algorithm is a very effective method for optimization, especially for largescale problems, because of its low memory requirement and simplicity. A modified scaled memoryless bfgs preconditioned conjugate. The parallel implementation of conjugate gradient linear system solver that i programmed here is designed to be used to solve large sparse systems of linear equations where the direct methods can exceed available machine memory andor be extremely timeconsuming. A new modified threeterm hestenesstiefel conjugate gradient. A modified prp conjugate gradient method a modified prp conjugate gradient method yuan, gonglin. A nonmonotone scaled conjugate gradient algorithm for largescale. A conjugate gradient method with guaranteed descent recently, a new nonlinear conjugate gradient scheme was developed which satisfies the descent condition gtkdk. A new accelerated conjugate gradient method for large. A new subspace minimization conjugate gradient method with. This paper proposes a new class of accelerated conjugate gradient like algorithms for solving large scale unconstrained optimization problems, which combine the idea of accelerated adaptive perry conjugate gradient algorithms proposed by andrei 2017 with the modified secant condition and the nonmonotone line search technique. Our modified threeterm conjugate gradient method possesses a sufficient descent property.
The proposed method is also equipped with a relaxed nonmonotone line search technique. The method combines the rivaiemustafaismail leong conjugate gradient method for unconstrained optimisation problems and a new nonmonotone linesearch method. Jul, 2006 2016 a modified prp conjugate gradient algorithm with nonmonotone line search for nonsmooth convex optimization problems. We suggest a conjugate gradient cg method for solving symmetric systems of nonlinear equations without computing jacobian and gradient via the special structure of the underlying function. We propose a nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization. It combines the steepest descent method with the famous conjugate gradient algorithm, which utilizes both the relevant function trait and the current point feature. A spectral conjugate gradient method under modified nonmonotone. The original hs method is the earliest conjugate gradient method.
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