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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 \(\min_{x\in \mathbb R^n} f(x)\). 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 \(N=\{x \in \mathbb R^n: f(x) \leq f(x_1)\}\) and \(f''(x)\) is Lipschitz continuous on \(N\). The boundedness of \(N\) is not assumed.

MSC:
65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C53 Methods of quasi-Newton type
Software:
CG_DESCENT; CUTEr; L-BFGS
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