A New Modified Conjugate Gradient Method and Its Global Convergence Theorem

Conjugate gradient methods are an important class of methods for solving unconstrained optimization problems, especially for large-scale problems. Recently, they have been much studied. In this paper, we further study the conjugate gradient method for unconstrained optimization. We focus our attention to the descent conjugate gradient method. This paper presents a modified conjugate gradient method. An interesting feature of the presented method is that the direction is always a descent direction for the objective function. Moreover, the property is independent of the line search used. Under mild conditions, we prove that the modified conjugate gradient method with Armijo-type line search is globally convergent. We also present some numerical results to show the efficiency of the proposed method. DOI: http://dx.doi.org/10.11591/telkomnika.v11i11.2894

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.

The conjugate gradient methods are numerously used for solving nonlinear unconstrained optimization problems, especially of large scale. Their wide applications are due to their simplicity and low memory requirement. To analyze conjugate gradient methods, two types of line searches are used; exact and inexact. In this paper, we present a new method of nonlinear conjugate gradient methods under the exact line search. The theoretical analysis shows that the new method generates a descent direction in each iteration and globally convergent under the exact line search. Moreover, numerical experiments based on comparing the new method with other well known conjugate gradient methods show that the new is efficient for some unconstrained optimization problems.

The most well-known technique or method in unconstrained optimization is the conjugate gradient method. It is used to get the greatest solution for the unconstrained optimization problems. This method is used in many fields especially, computer science, and engineering due to its convergence speed, simplicity, and the low memory requirements. A new modified conjugate gradient method is presented in this paper. This method is proved with the strong Wolfe-Powell (SWP) line search that it possesses sufficient descent property, and is globally convergent. Numerical results for a set of 141 test problems show the outperformance of the new proposed formula comparing with other methods using the same line search. The performance of this method is more efficient and better than the others.

Conjugate gradient (CG) method is a line search algorithm mostly known for its wide application in solving unconstrained optimization problems. Its low memory requirements and global convergence properties makes it one of the most preferred method in real life application such as in engineering and business. In this paper, we present a new CG method based on AMR* and CD method for solving unconstrained optimization functions. The resulting algorithm is proven to have both the sufficient descent and global convergence properties under inexact line search. Numerical tests are conducted to assess the effectiveness of the new method in comparison to some previous CG methods. The results obtained indicate that our method is indeed superior.

Nonlinear conjugate gradient methods for unconstrained optimization problems are used in many aspects of theoretical and applied sciences. They are iterative methods, so at any iteration a step length is computed using a method called line search. In most cases, the sufficient descent condition plays an important role to prove the global convergence of a conjugate gradient method. Due to its outperformance in practical computation, the Polak-Ribière-Polyak (PRP) conjugate gradient method is widely used for solving nonlinear unconstrained optimization problems. However, the sufficient descent condition of PRP has not established without line search yet. In this study, we established the sufficient descent condition without line search based on the conditions 0 < bkPRP £ x bkFR and 1 < bkPRP £mbkFR, where 0 < x < 1 and m > 1. As a result, we found that under certain conditions, the sufficient descent condition is satisfied when the PRP implemented without line search

At present, the conjugate gradient (CG) method of Hager and Zhang (Hager and Zhang, SIAM Journal on Optimization, 16(2005)) is regarded as one of the most effective CG methods for optimization problems. In order to further study the CG method, we develop the Hager and Zhang's CG method and present two modified CG formulas, where the given formulas possess the value information of not only the gradient but also the function. Moreover, the sufficient descent condition will be holden without any line search. The global convergence is established for nonconvex function under suitable conditions. Numerical results show that the proposed methods are competitive to the normal conjugate gradient method.

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.

The conjugate gradient (CG) method is one of the most popular methods to solve nonlinear unconstrained optimization problems. The Hestenes-Stiefel (HS) CG formula is considered one of the most efficient methods developed in this century. In addition, the HS coefficient is related to the conjugacy condition regardless of the line search method used. However, the HS parameter may not satisfy the global convergence properties of the CG method with the Wolfe-Powell line search if the descent condition is not satisfied. In this paper, we use the original HS CG formula with a mild condition to construct a CG method with restart using the negative gradient. The convergence and descent properties with the strong Wolfe-Powell (SWP) and weak Wolfe-Powell (WWP) line searches are established. Using this condition, we guarantee that the HS formula is non-negative, its value is restricted, and the number of restarts is not too high. Numerical computations with the SWP line search and some standard optimization problems demonstrate the robustness and efficiency of the new version of the CG parameter in comparison with the latest and classical CG formulas. An example is used to describe the benefit of using different initial points to obtain different solutions for multimodal optimization functions.

A popular optimization technique called the conjugate gradient method (CGM) is renowned for its effectiveness in addressing problems involving unconstrained optimization. Several conjugate gradient (CG) techniques have been proven to possess global convergence properties both theoretically and numerically. The Dai-Liao-type CGM is a variant that incorporates certain modifications to enhance its convergence properties. This paper examines the global convergence properties of a Dai-Liao-type CGM for unconstrained optimization problems. Theoretically, this study investigates the conditions under which the method ensures convergence to the global minimum of the objective function, focusing on the algorithm’s descent directions, the necessary reduction in objective function values, and termination criteria. A numerical experiment is conducted on a set of unconstrained optimization problems to validate the theoretical results obtained in this work. The numerical findings of this study demonstrate the robustness and reliability of the Dai-Liao-type CGM, showing its ability to find the global optimal solution in a wide range of unconstrained optimization problems.

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

In Andrei [Scaled memoryless BFGS preconditioned conjugate gradient (CG) algorithm for unconstrained optimization, Optim. Meth. Softw. 22(4) (2007), pp. 561–571], an efficient CG algorithm has been proposed for solving unconstrained optimization problems. However, due to a wrong inequality used in Andrei to show the sufficient descent property for the search directions, the proof of Theorem 2, the global convergence theorem, is incorrect. In what follows, the necessary corrections will be mentioned. Throughout, we use the same notations as in Andrei.

This study introduces an improved three-term conjugate gradient (CG) algorithm which fulfills the descent conditions and exhibits good global convergence properties. The development of this new algorithm is based on the findings of recently introduced generalized RMIL CG technique. The algorithm has been modified so that the search direction would always meet the sufficient descent condition no matter the line-search techniques employed. Under some given assumptions, the algorithm’s global convergence result for the general nonconvex functions is established. The numerical efficacy of the suggested three-term CG algorithm is assessed through comparisons with four other CG algorithms. The assessment is performed using an array of unconstrained optimization benchmark functions. The acquired findings indicate that all the considered algorithms exhibit similar performance, particularly the variants of the RMIL method. One algorithm is more robust and slightly faster than the rest. However, it is essential to highlight that the newly developed CG formula outperforms all the other algorithms, including a recently presented three-term PRP algorithm. The study further expanded the proposed method to address an image restoration problem. The computational experiments yielded promising results, proving that the proposed CG algorithm surpasses others. It produces higher-quality output images, requires less CPU time (sec), and achieves higher PSNR values.

Abstract: The conjugate gradient (CG) method is a one of the common approaches for solving unconstrained optimization problems, notably known for its suitability for large scale problems. Many recent studies show that this method is also useful for problems of smaller scale. One of the methods used for improving the performance of CG method is hybrid approach, where a CG method is combined with another method. In this study, the ARM CG method is combined with the SMR CG method and tested under exact line search. The resulting hybrid algorithm is globally convergent under exact line search and shown to perform well numerically in comparison to other tested CG methods.

To find a solution of unconstrained optimization problems, we normally use a conjugate gradient (CG) method since it does not cost memory or storage of second derivative like Newton’s method or Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. Recently, a new modification of Polak and Ribiere method was proposed with new restart condition to give a so-call AZPRP method. In this paper, we propose a new modification of AZPRP CG method to solve large-scale unconstrained optimization problems based on a modification of restart condition. The new parameter satisfies the descent property and the global convergence analysis with the strong Wolfe-Powell line search. The numerical results prove that the new CG method is strongly aggressive compared with CG_Descent method. The comparisons are made under a set of more than 140 standard functions from the CUTEst library. The comparison includes number of iterations and CPU time.