OPTEC - K.U.Leuven Center of Excellence: Optimization in Engineering

"Some recent research on minimization without derivatives"
M.J.D. Powell  (University of Cambridge)

The NEWUOA software for unconstrained minimization without derivatives
employs quadratic models of the objective function, F say. Each model
has (n+1)(n+2)/2 parameters, where n is the number of variables, but,
when solving some test problems, the total number of calculations of
F is only of magnitude n. This mystery will be explained. Each new model
is derived from the old model and about 2n+1 interpolation conditions,
the amount of routine computation of each iteration being only O(nn).
Thus several functions of 320 variables have been minimized successfully.
Recent research to be mentioned addresses moves of interpolation points
that are far from the best point so far, the number of interpolation
conditions, and the inclusion of prescribed lower and upper bounds on
the variables.