解奇异无约束优化问题的改进张量法

A Modified Tensor Method for Singular Unconstrained Optimization

给出一个解奇异无约束优化问题(极小点的Hessian矩阵奇异)的改进张量法。张量方法是标准牛顿模型方法的推广,它扩充目标函数的Taylor展式到四阶项,弥补了牛顿模型在极小点处的Hessian矩阵奇异时失去快速收敛性的缺陷。与标准张量法相比,本文主要的改进是,用梯度和二阶导数的差来替代函数与梯度差来构造张量模型。8个标准函数被奇异化后进行了数值试验,数值试验结果表明这个改进张量法是有效的。

In this paper, we propose a modified tensor method for singular unconstrained optimization where the Hessian is singular at the minimum point. The tensor model, which is a generalization of the standard Newton model and the extension to four-order term of the Taylor expansion, fix up the weakness that the Newton model will lose the fast local convergence rate of the standard Newton method where the Hessian is singular at the minimizer. Rather than with the difference of functions and gradients, the mod- ified tensor model is constructed with the difference of gradients and Hessian. We do the numerical experiments on eight standard test functions after singularizing. The numerical results show that the modified tensor method is effective.