A General Hybrid Optimization Strategy for Curve Fitting in the Non-uniform Rational Basis Spline Framework

In this paper, a general methodology to approximate sets of data points through Non-uniform Rational Basis Spline (NURBS) curves is provided. The proposed approach aims at integrating and optimizing the full set of design variables (both integer and continuous) defining the shape of the NURBS curve. To this purpose, a new formulation of the curve fitting problem is required: it is stated in the form of a constrained nonlinear programming problem by introducing a suitable constraint on the curvature of the curve. In addition, the resulting optimization problem is defined over a domain having variable dimension, wherein both the number and the value of the design variables are optimized. To deal with this class of constrained nonlinear programming problems, a global optimization hybrid tool has been employed. The optimization procedure is split in two steps: firstly, an improved genetic algorithm optimizes both the value and the number of design variables by means of a two-level Darwinian strategy allowing the simultaneous evolution of individuals and species; secondly, the optimum solution provided by the genetic algorithm constitutes the initial guess for the subsequent gradient-based optimization, which aims at improving the accuracy of the fitting curve. The effectiveness of the proposed methodology is proven through some mathematical benchmarks as well as a real-world engineering problem.