## 1 Introduction  
The high-precision sphere constitutes a pivotal component within the realm of precision instruments. A sphere's primary form tolerance characteristic manifests as a sphericity error. This inherent sphericity error, in turn, introduces unwelcome vibrations and noise, directly exerting a tangible influence on these precision instruments' precision and service longevity [1; 2]. The measurement and evaluation of the sphericity error are primarily conducted using three-coordinate measuring machines (CMMs) [3; 4]. This advanced technology enables the acquisition of extensive contour data for the spheres under examination, which are then meticulously stored in the XYZ coordinate format, forming a substantial dataset [5]. It is crucial to underscore that this dataset's precise and efficient processing stands as a critical imperative for the accurate evaluation of the sphericity error in spherical objects. According to the standards set by the American Society of Mechanical Engineers (ASME) and the International Organization for Standardization (ISO) [6; 7], the processingof datasets can be executed by applying four distinct criteria: the least squares criterion, the minimum circumscribed criterion, the maximum inscribed criterion, and the minimum zone criterion. The minimum zone criterion is the general default criterion for evaluating the sphericity error [8; 9; 10]. The aforementioned criterion enables the acquisition of a unique minimum sphericity error, and it is also the arbitration method when there is inconsistency in the evaluation of the sphericity error. The minimum zone criterion is an unconstrained and non-differentiable optimization problem, which poses significant challenges in terms of finding a viable solution [11; 12; 13]. Although the ASME and ISO standards stipulate that the evaluation of sampling point determination should be based on the minimum zone criterion, these standards do not provide a specific computational method for evaluating the spherical error based on the minimum zone criterion. Therefore, the development of a highly accurate and robust method for evaluating sphericity errors is of utmost urgency. Many scholars have extensively examined this issue by presenting a range of computational geometric methods [8; 14; 15; 16; 17; 18], region search techniques [19], heuristic algorithms [20; 21; 22], multi-objective optimization methods [23; 24; 25], and bio-inspired intelligent optimization algorithms [26; 27; 28; 29; 30] to address the problem. Among these approaches, region search methods are subject to inherent limitations that hinder their ability to accurately determine solutions for sphericity errors. Computational geometric methods for sampling point filtration exhibit enhanced search efficiency and improved results when dealing with fewer sampling points. Nevertheless, as the number of sampling points increases, the search cost escalates rapidly, resulting in unpredictable and fluctuating results. Heuristic algorithms are prone to being influenced by local optima and often struggle to converge effectively. Bio-inspired intelligent optimization algorithms yield more accurate results. However, they involve complex computations and numerous parameters that require optimization. Moreover, a sphericity error assessment, which is carried out to determine the optimal sphere center, minimizing the radius difference of concentric spheres that encompass the sampled contour, is fundamentally a single-objective optimization problem. Despite the potential advantages of applying multi-objective optimization methods to this evaluation, there is a paucity of scholars incorporating them into sphericity error assessment. To address the problem of inadequate precision in sphericity error evaluation methods, we conducted an in-depth examination of the spatial distribution characteristics of sphericity errors. Based on our research, it is recommended to approach the issue of sphericity errors in the local solution space by treating it as a problem of optimizing a unimodal function.