Performance Analysis of Steepest Descent-Line Search Condition Combinations in Nonlinear Least Squares Fitting of CMM Data

Abstract

This paper presents a benchmarking study on the steepest descent (SD) method considering three different line search conditions including Backtracking (BC), Armijo-Backtracking (ABC) and Goldstein (GC) in nonlinear least squares fitting of measured data obtained from coordinate measuring machine (CMM). Within this scope, five primitive geometries such as circle, square, rectangle, triangle and ellipse were built via 3D printer. Those geometries were then scanned with CMM to acquire their 2D profiles to be fitted. To find best fitting parameters for each geometry, the nonlinear least squares approach along with the above-mentioned optimization method-line search condition combinations were employed. During the fitting process, the total number of function evaluations, when the combination converges to required tolerance, were used as a performance metric of the combination in question. With those data, the performance and data profiles for each combination were created to be able to carry out a reliable performance evaluation. Based on those profiles, it has been seen that the SD-ABC combination is the fastest one. In addition, it is successful on all the geometries while the others are not. For the second fastest combination, the SD-BC combination stands out. However, its successful rate is only 80%, which means it fails on a geometry. On the other hand, the SD-GC combination takes the last place in this study. All those results have shown that the line search conditions have a great contribution to the success and performance of the optimization algorithm being used. Besides, their performance may differ from problem-to-problem. The end-users should consider these facts to find best optimization method-line search condition combination for their problems.