基于信赖域方法的几何约束求解技术的研究

The Research Based on the Trust Region Method Algorithm in the Geometric Constraint Solving

几何约束求解问题是当前基于约束设计研究中的热点问题。一个约束描述了一个应该被满足的关系,一旦用户已经定义了一系列的关系,那么在修改参数之后,系统会自动选择合适的状态来满足约束。拟将信赖域方法引入到几何约束求解中。因为传统的Newton法在实际计算时对初始点要求比较严格,且每次都要计算导数,当导数值出现奇异状况或非常小时,使计算无法进行,且收敛性不能保证,因而使方法受到一定的限制。信赖域方法既具有New-ton法的快速收敛性又有理想的总体收敛性,而且可以解决Hessian阵不正定和鞍点等困难。

The geometric constraint solving is a popular problem in the current constraint design research. A constraint can describe a relation to be satisfied. Once the user defines a series of relations, the system will select a proper state to satisfy the constraints after the parameters are modified. We will introduce the trust region method in the geometric constraint solving. Because traditional Newton method is strict to the initial point in actual computation, and each time we must calculate derivative every time. When the derivative value has the strange condition or is very smalll, it will cause the computation to be unable to be carried on. And the constringency cannot be ensured, thus the method is to limited in a certain. The trust region method not only has the fast astringency of the Newton method but also has a perfect overall contringency, moreover it may solve the difficulties of Hessian matrix non-positive definite and saddle point.