Subdivision Termination Criteria in Subdivision Multivariate Solvers
The need for robust solutions for sets of non-linear multivariate constraints or equations needs no motivation. Subdivision-based multivariate constraint solvers [1, 2, 3] typically employ the convex hull and subdivision/domain clipping properties of the Bézier/B-spline representation to detect all regions that may contain a feasible solution. Once such a region has been identified, a numerical improvement method is usually applied, which quickly converges to the root. Termination criteria for this subdivision/domain clipping approach are necessary so that, for example, no two roots reside in the same sub-domain (root isolation).
This work presents two such termination criteria. The first theoretical criterion identifies sub-domains with at most a single solution. This criterion is based on the analysis of the normal cones of the multiviarates and has been known for some time . Yet, a computationally tractable algorithm to examine this criterion has never been proposed. In this paper, we present such an algorithm for identifying sub-domains with at most a single solution that is based on a dual representation of the normal cones as parallel hyper-planes over the unit hyper-sphere. Further, we also offer a second termination criterion, based on the representation of bounding parallel hyper-plane pairs, to identify and reject sub-domains that contain no solution.
We implemented both algorithms in the multivariate solver of the IRIT  solid modeling system and present examples using our implementation.
KeywordsDual Representation Normal Cone Single Solution Geometric Design Tangent Cone
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