一个共轭梯度方法的全局收敛性

The Global Convergence of a Conjugate Gradient Method

关键词:  无约束优化  共轭梯度法  线搜索  全局收敛性

Key words:  unconstrained optimization  conjugate gradient method  line search  global convergence

摘要:

给出一种较弱的线搜索:寻找一个步长t_k=ρ^j_kΔ_k满足f(x_k+ρ^jd_k)-f(x_k)≤ αρ^jg_k^Td_k-m/2(ρ^j)²‖d_k‖²,α∈(0,1),ρ∈(0,1),m>0和g_k+1^Td_k+1<0,将此线搜索应用于求解共轭梯度公式的β_k^*,得到一种新共轭梯度算法,并证明新算法具有全局收敛,用数值实验说明新算法是有效的.

Abstract:

A line search is proposed.Under some suitable weak condition, we search a stepsize t_k=ρ^j_kΔ_k satisfying f (x_k+ρ^jd_k)-f (x_k) ≤ αρ^jg_k^Td_k-m/2 (ρ^j)²‖d_k‖², α∈ (0, 1), ρ∈ (0, 1), m>0 and g_k+1^Td_k+1<0.A new algorithm is obtained by using the present line search, and its global convergence is proved.Preliminary numerical results show that this method is efficient.