The conjugate gradient method is a mathematical technique that can be useful for the optimization of both linear and nonlinear systems. Pdf convergence conditions, line search algorithms and trust. Wong, 1970, computational methods in optimization 1971, and optimization. It works when the function is approximately quadratic near the minimum, which is the case when the function is twice differentiable at the minimum and the second derivative is nonsingular there. Gradient, newton, polak ribiere, bfgs methods and simplex method for optimization without the gradient rosenbrock test function is used to assess their performance. May 22, 2014 in order to achieve a theoretically effective and numerically efficient method for solving largescale unconstrained optimization problems, a hybridization of the fletcherreeves and polakribierepolyak conjugate gradient methods is proposed. Our approach is suitable for solving a large class of optimization problems on a rectangle of r n or unconstrained problems.
Java example source code file conjugategradientformula. A way to guarantee the polak ribiere method to converge is by choosing maxthis restarts the method if the polak ribiere method becomes less than zero, and then a new search direction can be chosen4. An efficient modified polakribierepolyak conjugate gradient method with global convergence properties. More, benchmarking optimization software with profile. In the proof, the global convergence of the polakribierepolak algorithm. Sufficient descent polakribierepolyak conjugate gradient. 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. It works when the function is approximately quadratic near the minimum, which is the case when the function is twice differentiable.
The proposed method possesses some attractive properties. We develop a new modified polakribiere conjugate gradient method by considering a random perturbation. In the method, the hybridization parameter is computed such that the generated search directions approach to the search directions of the efficient. Software downloads center for optimization and statistical learning. Fletcherreeves 19, polakribierepolyak 20, steepest descent 14, polakribierepolyak constrained by fletcherreeves, hagerzhang 21, and daiyuan 22. On the other hand, the polakribiere formula is often faster when it does converge. In this paper, we consider the global convergence of the polakribierepolyak abbreviated prp conjugate gradient method for unconstrained optimization problems. A modified polakribierepolyak conjugate gradient algorithm for largescale optimization problems. It also describes several packages developed during the last ten years, and illustrates their performance on some practical problems. In the present software implementation more efficient methods are provided, among which the fletcherreeves and polak ribiere conjugate gradient optimization methods. Although this condition is not needed in the convergence analyses of newton and quasinewton methods, gilbert and nocedal hint that the sufficient. The conjugate gradient cg method is one of the most popular methods for solving largescale unconstrained optimization problems. In this paper, we develop a threeterm polakribierepolyak conjugate gradient method, in which the search direction is close to the direction in the memoryless bfgs quasinewton method.
A hybridization of the polakribierepolyak and fletcher. In the present software implementation more efficient methods are provided, among which the fletcherreeves and polakribiere conjugate gradient optimization methods. This routine uses the fletcherreevespolakribiere method to approximately locate a local minimum of the usersupplied function fx. Recently, many modern applications of optimization have called for the need of. Three modified polakribierepolyak conjugate gradient. On the one hand, the fletcherreeves formula is guaranteed to converge if the start point is close enough of the optimum whether the polakribiere formula may not converge in rare cases. An efficient modified polak ribierepolyak conjugate gradient method with global convergence properties.
Cullum, 1970, notes for a first course on linear systems with e. Performance comparison of optimization methods for blind deconvolution daniel thompson the boeing company 550 lipoa parkway. A globally convergent version of the polakribiere gradient. Under appropriate conditions, the modified method is proved to possess global convergence under the wolfe or armijo. A hybrid conjugate gradient method for optimization problems. Combining the rosen gradient projection method with the twoterm polakribierepolyak prp conjugate gradient method, we propose a twoterm polakribierepolyak prp conjugate gradient projection method for solving linear equality constraints optimization problems. Pdf we study globally convergent implementations of the polakribiere pr. The gradient is calculated using the usersupplied function dfx, dfa where dfa is the ndimensional array whose i th component dfai is. Numerical comparisons with conjugate gradient algorithms using a set of 750 unconstrained optimization problems, some of them from the cute library, show that this computational scheme outperforms the known polakribierepolyak algorithm, as well as some other unconstrained optimization algorithms. In this paper, three modified polakribierepolyak prp conjugate gradient methods for unconstrained optimization are proposed.
An efficient modified polakribierepolyak conjugate gradient. Two new prp conjugate algorithms are proposed in this paper based on two modified prp conjugate gradient methods. Based on an eigenvalue analysis, a descent class of twoparameter extension of the conjugate gradient method proposed by polak and ribiere 1969, and polyak 1969 is suggested. A new generalized polakribiere conjugate gradient algorithm is proposed for unconstrained optimization, and its numerical and theoretical properties are discussed. It is difficult to predict which algorithm will perform best on a. A practical algorithm for solving largescale boxconstrained optimization problems is developed, analyzed, and tested. Two new prp conjugate gradient algorithms for minimization. Performance comparison of optimization methods for blind. A modified descent polakribierepolyak conjugate gradient. They are based on the twoterm prp method proposed by cheng numer. The new method is, in fact, a particular type of twodimensional newton method and is based on a finitedifference approximation to the product of a hessian and a vector. A new generalized polakribiere conjugate gradient algorithm is proposed for unconstrained optimization, and its numerical and theoretical properties the new method is, in fact, a particular type of twodimensional newton method and is based on a finitedifference approximation to the product of a hessian and a vector. Convergence of the polakribierepolyak conjugate gradient.
In particular, the socalled polakribierepolyak prp conjugate gradient method has. The new scheme reduces to the standard polakribierepolyak method when an exact line search is used. Optimization software, acm transactions on mathematical software, vol. Polakribieretype conjugate gradient listed as prcg. A modified polakribierepolyak conjugate gradient algorithm for. Nonlinear conjugate gradient methods with sufficient descent. Gradient, newton, polakribiere, bfgs methods and simplex method for optimization without the gradient rosenbrock test function is used to assess their performance. Pdf a modified polakribierepolyak conjugate gradient. Theory of mathematical programming and optimal control with m.
Software and user interfaces together with a simple numerical example and some more practical examples are described for the guidance of the user. In numerical optimization, the nonlinear conjugate gradient method generalizes the conjugate. By considering standard test problems, the superiority of the proposed software over some readily available library software and over the straightforward polakribiere algorithm is shown. In conjugate gradient methods scheme, polakribierepolyak prp method is. See or for a discussion of the polakribiere conjugate gradient algorithm. Convergence conditions, line search algorithms and trust region.
In the proposed algorithm, an identification strategy is. Fletcherreeves 19, polak ribiere polyak 20, steepest descent 14, polak ribiere polyak constrained by fletcherreeves, hagerzhang 21, and daiyuan 22. We study globally convergent implementations of the polakribiere pr conjugate gradient method for the. This article presents a modified quadratic hybridization of the polakribierepolyak and fletcherreeves conjugate gradient method for solving unconstrained optimization problems. 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 proposed method always generates a sufficient descent direction independent of the accuracy of the line search and the convexity of the objective function. We show that the method is globally convergent with the wolfe line search conditions as well as the backtracking armijotype line search strategy proposed by grippo and lucidi, without convexity. In addition, only a lower dimensional quadratic program subproblem needs. Numerical comparisons with conjugate gradient algorithms using a set of 750 unconstrained optimization problems, some of them from the cute library, show that this computational scheme outperform the known polakribierepolyak algorithm, as well as some other unconstrained optimization algorithms.
Numerical testing suggests that the polakribiere method tends to be more efficient than the fletcherreeves method. In this paper, a modification to the polakribierepolyak prp nonlinear conjugate gradient method is presented. 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. A three term polakribierepolyak conjugate gradient. A polakribierepolyak method for solving largescale nonlinear systems of equations and its global convergence. A new generalized polakribiere conjugate gradient algorithm is proposed for. The global proof of the polakribierepolak algorithm under the.
Global convergence, with the strong wolfe line search conditions, of the proposed quadratic hybrid conjugate gradient method is established. A modified sufficient descent polakribierepolyak type. A descent extension of the polakribierepolyak conjugate. We consider the boxconstrained optimization problem where is. A theoretical and computational investigation of a. Imsl, nagfortran, nagc, optima library, optpack, and proc nlp contain nonlinear conjugate gradient codes. A globally convergent version of the polakribiere gradient method. Siam journal on optimization society for industrial and. However, the usage of cg methods is mainly restricted to solving smooth optimization problems so far. A modified polak ribiere polyak conjugate gradient algorithm which satisfies both the sufficient descent condition and the conjugacy condition is presented. Qiglobal convergence of the polakribierepolyak conjugate gradient methods with inexact line search for nonconvex unconstrained optimization problems. Convergence properties of a correlative polakribiere conjugate gradient method. In this paper, we consider the unconstrained optimization problem where is a.
He is the author or coauthor of over 200 papers, as well as four books. Conjugate gradient backpropagation with polakribiere. A modified quadratic hybridization of polakribierepolyak and. Sufficient descent polakribierepolyak conjugate gradient algorithm for largescale boxconstrained optimization. Article pdf available in optimization methods and software 201.
Global optimization through a stochastic perturbation of. An international journal of optimization and control. In the proposed algorithm, an identification strategy is involved to estimate the active set at periteration. Theoretical results ensure that the proposed method converges to a global minimizer. The conjugate gradient cg method is one of the most popular methods for solving large scale unconstrained optimization problems. The new parameter provides global convergence properties when the. Contribute to qzhu2017optimization development by creating an account on github. A modified polakribierepolyak conjugate gradient algorithm for nonsmooth. In this paper, a new modified version of the cg formula that was introduced by polak, ribiere, and polyak is proposed for problems that are bounded below and have a lipschitzcontinuous gradient. Recently, important contributions on convergence studies of conjugate gradient methods were made by gilbert and nocedal siam j. By considering standard test problems, the superiority of the proposed software over some readily available library software and over the straightforward polak ribiere algorithm is shown. The simplest gradientbased optimization scheme is the steepest descent method. Extension of modified polakribierepolyak conjugate.
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