带扰动项的FR共轭梯度法(英文)

FR Conjugate Gradient Methods with Perturbations

本文提出了两种搜索方向带有扰动项的Fletcher-Reeves(abbr.FR)共轭梯度法.其迭代公式为x_k+1=x_k+α_k(s_k+ω_k),其中s_k由共轭梯度迭代公式确定,ω_k为扰动项,α_k采用线搜索确定而不是必须趋于零.我们在很一般的假设条件下证明了两种算法的全局收敛性,而不需要目标函数有下界或水平集有界等有界性条件.

In this paper, we propose two kinds of Fletcher-Reeves （abbr.FR） conjugate gradient methods with linesearch in the case that the search direction is perturbed slightly. Their iterate formula is xk＋1 = xk ＋ αk（sk ＋ ωk）, where the main direction sk is obtained by FR conjugate gradient method and ωk is perturbation term. The stepsize αk is determined by linesearch and needs not tend to zero. We prove that the two kinds of methods are globally convergent under mild conditions, and in doing so, we remove various boundedness conditions such as boundedness from blow of f, boundedness of level set, etc.