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Zbl 1093.90085
Hager, William W.; Zhang, Hongchao
A new conjugate gradient method with guaranteed descent and an efficient line search. (English)
[J] SIAM J. Optim. 16, No. 1, 170-192 (2005). ISSN 1052-6234; ISSN 1095-7189/e

Summary: A new nonlinear conjugate gradient method and an associated implementation, based on an inexact line search, are proposed and analyzed. With exact line search, our method reduces to a nonlinear version of the Hestenes--Stiefel conjugate gradient scheme. For any (inexact) line search, our scheme satisfies the descent condition ${\bold g}_k^{T} {\bold d}_k \le-\frac{7}{8}\|{\bold g}_k\|^2$. Moreover, a global convergence result is established when the line search fulfills the Wolfe conditions. A new line search scheme is developed that is efficient and highly accurate. Efficiency is achieved by exploiting properties of linear interpolants in a neighborhood of a local minimizer. High accuracy is achieved by using a convergence criterion, which we call the ``approximate Wolfe'' conditions, obtained by replacing the sufficient decrease criterion in the Wolfe conditions with an approximation that can be evaluated with greater precision in a neighborhood of a local minimum than the usual sufficient decrease criterion. Numerical comparisons are given with both L-BFGS and conjugate gradient methods using the unconstrained optimization problems in the CUTE library.
[Klaus Schittkowski (Bayreuth)]

MSC 2000:: *90C52 Methods of reduced gradient type
90C06 Large-scale problems

Keywords: conjugate gradient method; unconstrained optimization; convergence; line search; Wolfe conditions; nonlinear programming; global convergence; CUTE

Cited in: Zbl 1258.65056 Zbl 1258.65059 Zbl 1238.65052 Zbl 1228.90153 Zbl 1189.90221 Zbl 1154.90586 Zbl 1174.90892

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