圆度误差目标函数凸凹性的研究

应用凸函数理论证明了圆度误差最小区域评定法的目标函数是二维欧氏空间R2 中的连续、不可微的凸函数 ,从而证明了目标函数的全局极小值的唯一性 ,并给出了实例

Study on the Convex and Concave Character of Roundness Error Objective Function

The optimization algorithms are commonly used to approach the minima of the roundness objective functions through iteration when a microcomputer is applied to assess roundness errors by minimum zone, minimum circumscribed circle, and maximum inscribed circle methods. The essential prerequisite for convergence of any optimization algorithm is that the objective function to be solved has only one minimum in its definition domain, that is, it is a single valley one. If an objective function has more local minima in its definition domain , its solution searched for by an optimization algorithm may not be its global minimum which is the wanted roundness error. Therefore, the mathematical models and algorithms for roundness evaluation may be influenced in their solutions' reliability and practical value. By means of the theory of convex function , it is proved that the roundness objective function by minimum zone assessment is a continuous and non-differentiable and convex one defined in two-dimensional Euclidean space R 2. Therefore, the uniqueness of its global minimum is proved. Similar conclusion applies to the roundness objective functions by minimum circumscribed and maximum inscribed circle evaluation methods.