From Nonlinear Optimization to Convex Optimization through Firefly Algorithm and Indirect Approach with Applications to CAD/CAM

• Published in: TheScientificWorld Vol. 2013; no. 2013; pp. 1 - 10
• Main Authors: Gálvez, Akemi, Iglesias, Andrés
• Format: Journal Article
• Language: English
• Published: Cairo, Egypt Hindawi Publishing Corporation 01.01.2013; John Wiley & Sons, Inc; Wiley
• Subjects: Algorithms; Applied mathematics; Approximation; CAD; CAD/CAM; CAM; Computational geometry; Computer aided design; Computer aided manufacturing; Computer science; Continuity (mathematics); Convex analysis; Convex programming; Convexity; Curve fitting; Data points; Genetic algorithms; Genetic engineering; Heuristic methods; Knots; Manufacturing; Mathematical research; Methods; Nonlinear programming; Optimization; Optimization techniques; Parameterization; Singular value decomposition
• ISSN: 2356-6140; 1537-744X; 1537-744X
• DOI: 10.1155/2013/283919

Fitting spline curves to data points is a very important issue in many applied fields. It is also challenging, because these curves typically depend on many continuous variables in a highly interrelated nonlinear way. In general, it is not possible to compute these parameters analytically, so the problem is formulated as a continuous nonlinear optimization problem, for which traditional optimization techniques usually fail. This paper presents a new bioinspired method to tackle this issue. In this method, optimization is performed through a combination of two techniques. Firstly, we apply the indirect approach to the knots, in which they are not initially the subject of optimization but precomputed with a coarse approximation scheme. Secondly, a powerful bioinspired metaheuristic technique, the firefly algorithm, is applied to optimization of data parameterization; then, the knot vector is refined by using De Boor’s method, thus yielding a better approximation to the optimal knot vector. This scheme converts the original nonlinear continuous optimization problem into a convex optimization problem, solved by singular value decomposition. Our method is applied to some illustrative real-world examples from the CAD/CAM field. Our experimental results show that the proposed scheme can solve the original continuous nonlinear optimization problem very efficiently.