Spline and weighted random directions method for nonlinear optimization
For the unconstrained minimization problem \(\min \{F(x)| x\in {\mathbb{R}}^ n\}\) the following method is proposed: A random basis is generated and orthogonalized by the Gram-Schmidt algorithm to yield \(v_ 1,...,v_ n\). These directions are weighted to yield final search directions \(w_ i=W_ iv_ i\) with \(W_ i=diag(p_ 1,...,p_ n)\), \(p_ j=(x_ j^{(i)})^{-1/2}\) if \(x_ j^{(i)}>0\) and (for unknown reasons) simply \(p_ j=1\) if \(x_ j^{(i)}\leq 0\). In each direction of search 5 equidistant points are determined and a line search is based on either a cubic spline interpolate or a fourth degree interpolation polynomial. The algorithm does not become clear in all details from the paper. A proof of convergence or order of convergence is not given. Numerous numerical examples say little about the algorithm as they are not sufficiently documented (starting points, accuracy obtained) or compared with other algorithms. A reported number of 2310 function evaluations to minimize the 2-dimensional Rosenbrock function give little hope for a high efficiency.
scientific article; zbMATH DE number 3928233
scientific article