Developments of NEWUOA for minimization without derivatives

The NEWUOA software is described briefly, with some numerical results that show good efficiency and accuracy in the unconstrained minimization without derivatives of functions of up to 320 variables. Some preliminary work on an extension of NEWUOA that allows simple bounds on the variables is also described. It suggests a variation of a technique in NEWUOA for maintaining adequate linear independence in the interpolation conditions that are used, which leads to five versions of the technique including the original one. Numerical experiments suggest that the new versions have some merit, but the details of the calculations are influenced strongly by computer rounding errors. The dependence of the number of iterations on the number of interpolation conditions is also investigated numerically. A surprising case with n = 160 is found, n being the number of variables, where the number of iterations is reduced when the number of conditions is decreased from 2n + 1 to n + 6. The given conclusions may assist the development of some new software for unconstrained optimization.