From Nonlinear Optimization to Convex Optimization through Firefly Algorithm and Indirect Approach with Applications to CADCAM

الملخص EN

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.