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A nonlinear conjugate gradient method with a strong global convergence property. (English) Zbl 0957.65061
The paper presents a new version of the conjugate gradient method $x_{k+1}=x_k +\alpha_k d_k, d_{k+1}=-f''(x_{k+1})+ \beta_k d_k, \beta_k=\|f''(x_{k+1})\|^2/ d_k^T (f''(x_{k+1})-f''(x_k))$ for solving unconstrained optimization problem $\min_{x\in \Bbb R^n} f(x)$. Unlike the classical stepsize rule $\alpha_k =\text{argmin}\{ f(x_k+\alpha d_k): \alpha \geq 0 \}$ the authors analyse a scheme in which $\alpha_k$ is chosen arbitrarily subject to the conditions $f(x_k)-f(x_k+\alpha_k d_k) \geq -\delta \alpha_k f^{\prime T}(x_k) d_k$ and $f^{\prime T}(x_k+\alpha_k d_k) d_k > \sigma f^{\prime T}(x_k) d_k; 0<\delta <\sigma <1$. The convergence $f''(x_k) \to 0$ is established provided that $f(x)$ is bounded below on $N=\{x \in \Bbb R^n: f(x) \leq f(x_1)\}$ and $f''(x)$ is Lipschitz continuous on $N$. The boundedness of $N$ is not assumed.

65K05Mathematical programming (numerical methods)
90C30Nonlinear programming
90C53Methods of quasi-Newton type
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