A three-term conjugate gradient method under the strong-Wolfe line search

Conjugate gradient methods are a popular class of iterative methods for solving linear systems of equations and nonlinear optimization problems as they do not require the storage of any matrices. In order to obtain a theoretically effective and numerically efficient method, two modified conjugate gradient methods (called the MCB1 and MCB2 methods) are proposed. In which the coefficientβkin the two proposed methods is inspired by the structure of the conjugate gradient parameters in some existing conjugate gradient methods. Under the strong Wolfe line search, the sufficient descent property and global convergence of the MCB1 method are proved. Moreover, the MCB2 method generates a descent direction independently of any line search and produces good convergence properties when the strong Wolfe line search is employed. Preliminary numerical results show that the MCB1 and MCB2 methods are effective and robust in minimizing some unconstrained optimization problems and each of these modifications outperforms the four famous conjugate gradient methods. Furthermore, the proposed algorithms were extended to solve the problem of mode function.

In this paper, we present a new hybrid method to solve a nonlinear unconstrained optimization problem by using conjugate gradient, which is a convex combination of Liu–Storey (LS) conjugate gradient method and Hager–Zhang (HZ) conjugate gradient method. This method possesses the sufficient descent property with Strong Wolfe line search and the global convergence with the strong Wolfe line search. In the end of this paper, we illustrate our method by giving some numerical examples.

The stationary point of optimization problems can be obtained via conjugate gradient (CG) methods without the second derivative. Many researchers have used this method to solve applications in various fields, such as neural networks and image restoration. In this study, we construct a three-term CG method that fulfills convergence analysis and a descent property. Next, in the second term, we employ a Hestenses-Stiefel CG formula with some restrictions to be positive. The third term includes a negative gradient used as a search direction multiplied by an accelerating expression. We also provide some numerical results collected using a strong Wolfe line search with different sigma values over 166 optimization functions from the CUTEr library. The result shows the proposed approach is far more efficient than alternative prevalent CG methods regarding central processing unit (CPU) time, number of iterations, number of function evaluations, and gradient evaluations. Moreover, we present some applications for the proposed three-term search direction in image restoration, and we compare the results with well-known CG methods with respect to the number of iterations, CPU time, as well as root-mean-square error (RMSE). Finally, we present three applications in regression analysis, image restoration, and electrical engineering.

In this paper, we propose a three-term conjugate gradient method based on secant conditions for unconstrained optimization problems. Specifically, we apply the idea of Dai and Liao (in Appl. Math. Optim. 43: 87–101, 2001) to the three-term conjugate gradient method proposed by Narushima et al. (in SIAM J. Optim. 21: 212–230, 2011). Moreover, we derive a special-purpose three-term conjugate gradient method for a problem, whose objective function has a special structure, and apply it to nonlinear least squares problems. We prove the global convergence properties of the proposed methods. Finally, some numerical results are given to show the performance of our methods.

Recently, conjugate gradient methods, which usually generate descent search directions, are useful for large-scale optimization. Narushima et al. (SIAM J Optim 21:212---230, 2011) have proposed a three-term conjugate gradient method which satisfies a sufficient descent condition. We extend this method to two parameters family of three-term conjugate gradient methods which can be used to control the magnitude of the directional derivative. We show that these methods converge globally and work well for suitable choices of the parameters. Numerical results are also presented.

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.

The nonlinear conjugate gradient method (CGM) is a very effective iterative method for solving large-scale optimal problems. In this paper, based on a variant of Polak–Ribiere–Polyak method, two modified CGMs with disturbance factors are proposed. By the disturbance factors, the two proposed methods not only generate sufficient descent direction at each iteration but also converge globally for nonconvex minimization if the strong Wolfe line search is used. Finally, elementary numerical experiment results are reported, which show that the proposed methods are promising.

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. For any line search, the new method satisfies the sufficient descent condition . Moreover, we prove that the new method is globally convergent under the strong Wolfe line search. The numerical results show that the new method is more effective for the given test problems from the CUTE test problem library (Bongartz et al., 1995) in contrast to the famous CD method, FR method, and PRP method.

<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>

In this paper, we propose a three-parameter family of conjugate gradient methods for unconstrained optimization. The three-parameter family of methods not only includes the already existing six practical nonlinear conjugate gradient methods, but subsumes some other families of nonlinear conjugate gradient methods as its subfamilies. With Powell’s restart criterion, the three-parameter family of methods with the strong Wolfe line search is shown to ensure the descent property of each search direction. Some general convergence results are also established for the three-parameter family of methods. This paper can also be regarded as a brief review on nonlinear conjugate gradient methods.

Conjugate gradient methods are a class of very effective iterative methods for large-scale unconstrained optimization. In this paper, a new Dai-Liao conjugate gradient method for solving large-scale unconstrained optimization problem is proposed. Based on the approximately optimal stepsize for the gradient method, we derive three new choices for the important parameters tk in Dai-Liao conjugate gradient method. The search direction satisfies the sufficient descent condition, and the global convergences of the proposed method for uniformly convex and general functions are proved under some mild conditions. Numerical experiments on a set of test problems from the CUTEst library show that the proposed method is superior to some well-known conjugate gradient methods.

We developed a new three-term conjugate gradient method (C.G.M)in this study. Under certain conditions, this approach achieves the desired descent direction while also being globally convergent. On the basis of the Dolan-More performance profile, these results show that this method outperforms the alternative standard method

A modified Polak–Ribière–Polyak conjugate gradient algorithm for nonsmooth convex programs

In this article, we try to proposed a new conjugate gradient method for solving unconstrained optimization problems, we focus on conjugate gradient methods applied to the non-linear unconstrained optimization problems, the positive step size is obtained by a line search and the new scalar to the new direction for the conjugate gradient method is derived from the quadratic function and Taylor series and by using quasi newton condition and Newton direction while deriving the new formulae. We also prove that the search direction of the new conjugate gradient method satisfies the sufficient descent and all assumptions of the global convergence property are considered and proved .in order to complete the benefit of our research we should take into account studied the numerical results which are written in FORTRAN language when the objective function is compared our new algorithm with HS and PRP methods on the similar set of unconstrained optimization test problems which is very efficient and encouragement numerical results.

An accelerated descent CG algorithm with clustering the eigenvalues for large-scale nonconvex unconstrained optimization and its application in image restoration problems