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Another hybrid conjugate gradient algorithm for unconstrained optimization

Authors:	Neculai Andrei

Institution:	(1) Research Institute for Informatics, Center for Advanced Modeling and Optimization, 8-10, Averescu Avenue, Bucharest 1, Romania

Abstract:	Another hybrid conjugate gradient algorithm is subject to analysis. The parameter β _k is computed as a convex combination of $\beta ^{{HS}}_{k}$ (Hestenes-Stiefel) and $\beta ^{{DY}}_{k}$ (Dai-Yuan) algorithms, i.e. $\beta ^{C}_{k} = {\left( {1 - \theta _{k} } \right)}\beta ^{{HS}}_{k} + \theta _{k} \beta ^{{DY}}_{k}$ . The parameter θ _k in the convex combination is computed in such a way so that the direction corresponding to the conjugate gradient algorithm to be the Newton direction and the pair (s _k, y _k) to satisfy the quasi-Newton equation $\nabla ^{2} f{\left( {x_{{k + 1}} } \right)}s_{k} = y_{k}$ , where $s_{k} = x_{{k + 1}} - x_{k}$ and $y_{k} = g_{{k + 1}} - g_{k}$ . The algorithm uses the standard Wolfe line search conditions. Numerical comparisons with conjugate gradient algorithms show that this hybrid computational scheme outperforms the Hestenes-Stiefel and the Dai-Yuan conjugate gradient algorithms as well as the hybrid conjugate gradient algorithms of Dai and Yuan. A set of 750 unconstrained optimization problems are used, some of them from the CUTE library.

Keywords:	Unconstrained optimization Hybrid conjugate gradient method Newton direction Numerical comparisons
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