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

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 pr...

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Vydáno v:TheScientificWorld Ročník 2013; číslo 2013; s. 1 - 10
Hlavní autoři: Gálvez, Akemi, Iglesias, Andrés
Médium: Journal Article
Jazyk:angličtina
Vydáno: Cairo, Egypt Hindawi Publishing Corporation 01.01.2013
John Wiley & Sons, Inc
Wiley
Témata:
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
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Shrnutí: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.
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Academic Editors: Z. Cui and X. Yang
ISSN:2356-6140
1537-744X
1537-744X
DOI:10.1155/2013/283919