Abstract
One of the important methods that are widely utilized to resolve unconstrained optimization problems is the Conjugate Gradient (CG) method. This paper aims to propose a new version of the CG method on the basis of Weak Wolfe-Powell (WWP) line search. The assumption is bounded below optimization problems with the Lipschitz continuous gradient. The new parameter obtains global convergence properties when the WWP line search is used. The descent condition is established without using any line search. The performance of the proposed CG method is tested by obtaining some unconstrained optimization problems from the CUTEst library. Testing results show that the proposed version of the CG method outperforms CG-DESCENT version 5.3 in terms of CPU time, function evaluations, gradient evaluations and number of iterations.
Highlights
The Conjugate Gradient (CG) method is utilized to resolve unconstrained optimization problems in the form of:f x, x n, The search direction dk of the CG method is defined in Equation (2): ddkk gk gk k dk 1 if k 1 if k 2 (2)where f : n represents the smooth function and denotes that the gradient is available
The modified formula is restarted on the basis of the value of the Lipchitz constant
The global convergence is established by using Weak Wolfe-Powell (WWP)
Summary
Using the CG method does not require a second derivative or its approximation as Newtons method or its modifications By generating a sequence of points xk+1 (Equation (1), start from initial point x0, where xk denotes the current iteration and ak > 0 indicates a step length obtained from a line search (Equation (2)-(6). The exact line search, which is expressed in Equation (3), can be used to obtain the step length: min f xk dk , 0 (3). This type of line search is computationally expensive because numerous iterations are required to obtain the step length. The inexact line search is not as computationally expensive as
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