求解非凸优化问题的一类Broyden算法超线性收敛性分析(英文)

Superlinear Convergence Analysis of Broyden Methods for Solving Nonconvex Optimization Problems

提出了一类求解无约束优化问题的修正Broyden算法。当目标函数非凸时,在一般情况下它的二阶导数矩阵可能不正定,为有效地利用Broyden算法求解非凸优化问题,通过建立新的正割方程,设计了一类修正的Broyden算法.当假设目标函数二阶连续可微,目标函数的梯度和Hesse矩阵满足Lipschitz条件,线性搜索满足Wolfe原则时,讨论了所设计算法的超线性收敛性。

In this paper, a class of modified Broyden methods are proposed for solving unconstrained optimization problems. Since the Hessian matrix of objective function is generally not positive definite when the objective function is nonconvex, it would be reasonable to expect that a proper modification of the Broyden methods is effective for nonconvex problems. Based on this view, a new secant equation for the methods is given, and a calss of modified Broyden methods are presented. Furthermore, if the objective function is twice continuously differentiable, the gradient and Hessian matrix are Lipschitz continuous, the line search satis-fies a standard sufficient decrease condition, the superlinear convergence of the methods can be established.