Armijo线性搜索下Hager-Zhang共轭梯度法的全局收敛性

Hager和Zhang^4]提出了一种新的非线性共轭梯度法(简称 HZ 方法), 并证明了该方法在 Wolfe搜索和 Goldstein 搜索下求解强凸问题的全局收敛性.但是HZ方法在标准Armijo 搜索下求解非凸问题是否全局收敛尚不清楚.该文提出了一种保守的HZ共轭梯度法,并且证明了这种方法在 Armijo 线性搜索下求解非凸优化问题的全局收敛性.此外,作者给出了一些 数值结果以检验该方法的有效性.

On the Global Convergence of the Hager-Zhang Conjugate Gradient Method with Armijo Line Search

Hager and Zhang in 4] proposed a new nonlinear conjugate gradient method (HZ method) and proved that this method is globally convergent when the line search fulfills the Wolfe conditions or the Goldstein’s conditions for strongly convex functions. But no global convergence results were obtained for nonconvex objective functions with Armijo line search. In this paper, the authors introduce a cautious HZ method and prove that the proposed method with Armijo line search converges globally even if the minimization function is nonconvex. The authors also present some numerical results to show the efficiency of the proposed method.