A Hybrid Conjugate Gradient Algorithm Withmodified Secant Condition For unconstrained Optimization As A Convex Combination Of Hestenes-stiefel And Dai-yuan Algorithms

摘要

Another hybrid conjugate gradient algorithm is suggested in this paper. The parameter β_k is computed as a convex combination of β_k~(HS) (Hestenes-Stiefel) and β_k~(DY) (Dai-Yuan) formulae, i.e. β_k~C = (1 - θ_k )β_k~(HS) + θ_kβ_k~(DY). 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 modified secant condition given by Zhang et al. [32] and Zhang and Xu [33], 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 a variant of the hybrid conjugate gradient algorithm given by Andrei [6], in which the pair (s_k, y_k) satisfies the secant condition ▽~2 f(x_(k+1))s_k = y_k , as well as the Hestenes-Stiefel, the Dai-Yuan conjugate gradient algorithms, and 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.