Truncated Smoothing Newton Method for $l_\infty$ Fitting Rotated Cones

DOI：10.3770/j.issn:1000-341X.2010.01.015

 作者 单位 肖瑜 大连理工大学数学科学学院, 辽宁 大连 116024 于波 大连理工大学数学科学学院, 辽宁 大连 116024 王德伦 大连理工大学机械工程学院, 辽宁 大连 116024

旋转锥面$l_\infty$拟合问题出现在机械制造中的一些反向工程中,是一类特殊的约束非光滑优化问题,其目标函数是一些包含绝对值函数和根号函数的非光滑函数的极大值函数.尽管这是一个低维问题,但在实际应用中,比如轴和轴套自动匹配问题,需要对大批量锥形零件的测量数据进行旋转锥面拟合.另外,如果每个锥形零件的测量数据量很大,极大值函数的组成函数个数很大.因此,有必要研究该问题的更有效的解法.为了有效的求解此问题,我们提出了一种截断光滑化牛顿法.首先对max函数的组成函数光滑化,并采用凝聚函数对max函数光滑化,

In this paper, the rotated cone fitting problem is considered. In case the measured data are generally accurate and it is needed to fit the surface within expected error bound, it is more appropriate to use $l_\infty$ norm than $l_2$ norm. $l_\infty$ fitting rotated cones need to minimize, under some bound constraints, the maximum function of some nonsmooth functions involving both absolute value and square root functions. Although this is a low dimensional problem, in some practical application, it is needed to fitting large amount of cones repeatedly, moreover, when large amount of measured data are to be fitted to one rotated cone, the number of components in the maximum function is large. So it is necessary to develop efficient solution methods. To solve such optimization problems efficiently, a truncated smoothing Newton method is presented. At first, combining aggregate smoothing technique to the maximum function as well as the absolute value function and a smoothing function to the square root function, a monotonic and uniform smooth approximation to the objective function is constructed. Using the smooth approximation, a smoothing Newton method can be used to solve the problem. Then, to reduce the computation cost, a truncated aggregate smoothing technique is applied to give the truncated smoothing Newton method, such that only a small subset of component functions are aggregated in each iteration point and hence the computation cost is considerably reduced.