A Modified Conjugate Gradient Method for Unconstrained Optimization

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.

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 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.

<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>

A study of Liu-Storey conjugate gradient methods for vector optimization

The best spectral conjugate gradient algorithm by (Birgin, E. and Martínez, J.M., 2001, A spectral conjugate gradient method for unconstrained optimization. Applied Mathematics and Optimization, 43, 117–128). which is mainly a scaled variant of (Perry, J.M., 1977, A class of Conjugate gradient algorithms with a two step varaiable metric memory, Discussion Paper 269, Center for Mathematical Studies in Economics and Management Science, Northwestern University), is modified in such a way as to overcome the lack of positive definiteness of the matrix defining the search direction. This modification is based on the quasi-Newton BFGS updating formula. The computational scheme is embedded into the restart philosophy of Beale–Powell. The parameter scaling the gradient is selected as spectral gradient or in an anticipative way by means of a formula using the function values in two successive points. In very mild conditions it is shown that, for strongly convex functions, the algorithm is global convergent. Computational results and performance profiles for a set consisting of 700 unconstrained optimization problems show that this new scaled nonlinear conjugate gradient algorithm substantially outperforms known conjugate gradient methods including: the spectral conjugate gradient SCG by Birgin and Martínez, the scaled Fletcher and Reeves, the Polak and Ribière algorithms and the CONMIN by (Shanno, D.F. and Phua, K.H., 1976, Algorithm 500, Minimization of unconstrained multivariate functions. ACM Transactions on Mathematical Software, 2, 87–94).

Nonlinear conjugate gradient (CG) methods are widely used for solving large scale unconstrained optimization problems. Many studies have been devoted to modified and improve this method. In this paper, a new parameter of CG method that possesses global convergence properties using exact line search is proposed. Numerical results show that the new formula is best and more efficient when compared with the other classical CG methods

<p>The conjugate gradient methods are noted to be exceedingly valuable for solving large-scale unconstrained optimization problems since it needn't the storage of matrices. Mostly the parameter conjugate is the focus for conjugate gradient methods. The current paper proposes new methods of parameter of conjugate gradient type to solve problems of large-scale unconstrained optimization. A Hessian approximation in a diagonal matrix form on the basis of second and third-order Taylor series expansion was employed in this study. The sufficient descent property for the proposed algorithm are proved. The new method was converged globally. This new algorithm is found to be competitive to the algorithm of fletcher-reeves (FR) in a number of numerical experiments.</p>

The Spectral conjugate gradient (SCG) methods are among the efficient variants of CG algorithms which are obtained by combining the spectral gradient parameter and CG parameter. The success of SCG methods relies on effective choices of the step-size $\alpha _{k}$ and the search direction $d_{k}$ . This paper presents an SCG method for unconstrained optimization models. The search directions generated by the new method possess sufficient descent property without the restart condition and independent of the line search procedure used. The global convergence of the new method is proved under the weak Wolfe line search. Preliminary numerical results are presented which show that the method is efficient and promising, particularly for large-scale problems. Also, the method was applied to solve the robotic motion control problem and portfolio selection problem.

Many scientific applications require one to solve successively linear systems Ax=b with different right-hand sides b and a symmetric positive definite matrix A. The conjugate gradient method applied to the first system generates a Krylov subspace which can be efficiently recycled thanks to orthogonal projections in subsequent systems. A modified conjugate gradient method is then applied with a specific initial guess and initial descent direction and a modified descent direction during the iterations. This paper gives new theoretical results for this method and proposes a new version. Numerical experiments show the efficacy of our method even for quite different right-hand sides.

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.

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.

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.

Metaheuristic algorithms are used to solve many optimization problems. Firefly algorithm, particle swarm improvement, harmonic search, and bat algorithm are used as search algorithms to find the optimal solution to the problem field. In this paper, we have investigated and analyzed a new scaled conjugate gradient algorithm and its implementation, based on the exact Wolfe line search conditions and the restart Powell criterion. The new spectral conjugate gradient algorithm is a modification of the Birgin and Martínez method, a manner to overcome the lack of positive definiteness of the matrix defining the search direction. The preliminary computational results for a set of 30 unconstrained optimization test problems show that this new spectral conjugate gradient outperforms a standard conjugate gradient in this field and we have applied the newly proposed spectral conjugate gradient algorithm in bat algorithm to reach the lowest possible goal of bat algorithm. The newly proposed approach, namely, the directional bat algorithm (CG-BAT), has been then tested using several standard and nonstandard benchmarks from the CEC’2005 benchmark suite with five other algorithms and has been then tested using nonparametric statistical tests and the statistical test results show the superiority of the directional bat algorithm, and also we have adopted the performance profiles given by Dolan and More which show the superiority of the new algorithm (CG-BAT).

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.