This new line search technique is based on a relaxation of the strong wolfe conditions and it allows to accept larger steps. The result is conjugate gradient on the normal equations cgnr. 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. A class of nonmonotone conjugate gradient methods for. Pdf the limited memory conjugate gradient method semantic. The additional orthogonality reduces the gramschmidt process to threeterm recursion. This section establishes the multipreconditioned analogy of cg in a fashion similar to the derivation of the standard pcg, whose. Nonmonotone conjugate gradient methods for optimization. We applied the spectral steplength to the entire conjugate gradient direction rather than the negative gradient. In this paper, we introduce a class of nonmonotone conjugate gradient methods, which include the wellknown polakribire method and hestenesstiefel method as special cases. On the convergence of a new conjugate gradient algorithm. We introduce two novel vector transports associated with the retraction constructed by the cayley transform. With exact line search, our method reduces to a nonlinear ver. Request pdf the convergence of conjugate gradient method with nonmonotone line search the conjugate gradient method is a useful and powerful approach for solving largescale minimization problems.
In this paper, we propose a nonmonotone line search combining with the search direction g. In this paper, we combined a new nonmonotone techniques with the spectral conjugate gradient method to obtain a more e. In this paper, combining the nonmonotone and monotone line search,a spectral conjugate gradient methods are used in this paper. 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 nonmonotone conjugate gradient algorithm for unconstrained. The global convergence of the given method will be established under suitable condi. A modified spectral conjugate gradient methods with the new. 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. A new conjugate gradient method with guaranteed descent. An iterative conjugate gradient regularization method for. This class of nonmonotone conjugate gradient methods is proved to be globally convergent when it is applied to solve unconstrained optimization problems with convex objective functions.
A nonmonotone prp conjugate gradient method for solving. However, for some illconditioned problems, orthogonality is quickly lost due to rounding errors, and convergence is much slower than expected. 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. The new residual is orthogonal to the whole space not only to one residual vector in the previous step. Although the steepest descent method converges, it is inef. Image restoration is an illposed inverse problem, which has been introduced the regularization method to suppress over. In this paper, we introduce a class of nonmonotone conjugate gradient methods, which include the wellknown polakribiere method and hestenesstiefel method as special cases.
The convergence of conjugate gradient method with nonmonotone. A new nonmonotone spectral conjugate gradient method for. Under mild assumptions, we prove the global convergence and linear convergence rate of the method. Request pdf the convergence of conjugate gradient method with nonmonotone line search the conjugate gradient method is a useful and powerful approach for. The spectral gradient and conjugate gradient methods are a class of methods that can suitably cope with largescale settings.
Nonlinear conjugate gradient methods, unconstrained optimization, nonlinear. Nonmonotone spectral projected gradient methods on convex sets. Numerical experiments show that the nonmonotone polakribiere method and hestenesstiefel method in this nonmonotone conjugate gradient class are. In this paper conjugate gradient methods with nonmonotone line search technique are introduced. A nonmonotone hybrid conjugate gradient method for. Optimization online a riemannian conjugate gradient method. The spectral projected gradient spg method birgin, mart nez, and raydan2000,2001. This paper is devoted to a riemannian conjugate gradient method for solving problem 1. Nov 10, 2016 in this paper we propose a new riemannian conjugate gradient method for optimization on the stiefel manifold. Dais nonmonotone conjugate gradient method is generalized to the riemannian case and global convergence of the new algorithm is established under standard assumptions.
A nonmonotone line search method for regression analysis. The cg method has the simplicity and the very low memory requirement and the prp method is one of the most effective conjugate gradient methods. A scaled conjugate gradient method based on new bfgs secant. Therefore, there existed a wide space for variations and extensions of the bb original method. A new class of conjugate gradient methods with extended. Preliminary numerical results show that this method is very ef.
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. This makes them easy to implement and they do not require much storage. Abstract pdf 679 kb 2017 a nonmonotone prp conjugate gradient method for solving square and underdetermined systems of equations. A new conjugate gradient method with guaranteed descent and. The steepest descent and the conjugate gradient methods both do not require estimation of the hessian. When we write p 1, we never intend that an inverse will be explicitly computed. In particular, a scaled version of the conjugate gradient method, suggested by perry 2, 9, which employ the spectral steplength of barzilai and borwein 1, 10, was. A comparative study of non linear conjugate gradient methods.
By making use of the moreauyosida regularization, a nonmonotone line search technique of 48 and a new secant equation of 43 derived by the authors earlier, we present a modified prp conjugate. A limited memory version of the nonlinear conjugate gradient method is developed. A new nonlinear spectral conjugate descent method for solving unconstrained optimization problems is proposed on the basis of the cd method and the spectral conjugate gradient method. We then of n are being very large, say, n 106 or n 107. P may come from incomplete lu, or a few steps of a multigrid iteration, or. A nonmonotone line search technique for newtons method. A scaled conjugate gradient method based on new bfgs. A modified polakribierepolyak conjugate gradient algorithm. Both of them satisfy the ringwirth nonexpansive condition, which is fundamental for convergence analysis of riemannian conjugate gradient methods, and one of them is also. A nonmonotone hybrid conjugate gradient method is proposed, in which the technique of the nonmonotone wolfe line search is used.
On the subspace minimization conjugate gradient method yuhong dai center for optimization and applications amss, chinese academy of sciences discussing with y. An introduction to the conjugate gradient method without the. Numerical results on a variety of lowrank test problems demonstrate the effectiveness of the new method. Krylov subspaces and conjugate gradients c 2006 gilbert strang 6. Conjugate gradient method employs vectors that are aorthogonal or conjugate details of the derivation of the method are omitted r 1 r 0 0 t j 0 t d i ad. 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. A new nonlinear conjugate gradient method and an associated implementation, based on an inexact line search, are proposed and analyzed. A nonmonotone conjugate gradient algorithm for unconstrained optimization article in journal of systems science and complexity 152 january 2002 with 60 reads how we measure reads.
This lemma shows the advantage of the conjugate gradient method over the gradient method. A riemannian conjugate gradient method for optimization on. Nonmonotone conjugate gradient method nonmonotone line search global convergence unconstrained optimization. Wei, new line search methods for unconstrained optimization, journal of the korean statistical society, 382009, pp. In theory, the successive gradients generated by the conjugate gradient method applied to a quadratic should be orthogonal. It is well known that in the euclidean space, the conjugate gradient method generally outperforms the steepest descent method for its faster convergence and is more suitable than secondorder methods such as newtons method, quasinewton. A modi ed of nonmonotone spectral conjugate gradient method. The spectral gradient method has been successfully extended in for solving square nonlinear systems of equations using grippos nonmonotone line search technique.
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