# bobyqa

0th

Percentile

##### An R interface to the bobyqa implementation of Powell

The purpose of bobyqa is to minimize a function of many variables by a trust region method that forms quadratic models by interpolation. Box constraints (bounds) on the parameters are permitted.

Keywords
nonlinear, optimize
##### Usage
bobyqa(par, fn, lower = -Inf, upper = Inf, control = list(), ...)
##### Arguments
par
A numeric vector of starting estimates of the parameters of the objective function.
fn
A function that returns the value of the objective at the supplied set of parameters par using auxiliary data in .... The first argument of fn must be par.
lower
A numeric vector of lower bounds on the parameters. If the length is 1 the single lower bound is applied to all parameters.
upper
A numeric vector of upper bounds on the parameters. If the length is 1 the single upper bound is applied to all parameters.
control
An optional list of control settings. See the details section for the names of the settable control values and their effect.
...
Further arguments to be passed to fn.
##### Details

The function fn must return a scalar numeric value.

The control argument is a list. Possible named values in the list and their defaults are:

npt
The number of points used to approximate the objective function via a quadratic approximation. The value of npt must be in the interval $[n+2,(n+1)(n+2)/2]$ where $n$ is the number of parameters in par. Choices that exceed $2*n+1$ are not recommended. If not defined, it will be set to $min(n * 2, n+2)$.

rhobeg
rhobeg and rhoend must be set to the initial and final values of a trust region radius, so both must be positive with 0 < rhoend < rhobeg. Typically rhobeg should be about one tenth of the greatest expected change to a variable. If the user does not provide a value, this will be set to min(0.95, 0.2 * max(abs(par))). Note also that smallest difference abs(upper-lower) should be greater than or equal to rhobeg*2. If this is not the case then rhobeg will be adjusted.

rhoend
The smallest value of the trust region radius that is allowed. If not defined, then 1e-6 times the value set for rhobeg will be used.

iprint
The value of iprint should be set to an integer value in 0, 1, 2, 3, ..., which controls the amount of printing. Specifically, there is no output if iprint=0 and there is output only at the start and the return if iprint=1. Otherwise, each new value of rho is printed, with the best vector of variables so far and the corresponding value of the objective function. Further, each new value of the objective function with its variables are output if iprint=3. If iprint > 3, the objective function value and corresponding variables are output every iprint evaluations. Default value is 0.

maxfun
The maximum allowed number of function evaluations. If this is exceeded, the method will terminate.

##### Value

A list with components:
par
The best set of parameters found.
fval
The value of the objective at the best set of parameters found.
feval
The number of function evaluations used.
ierr
An integer error code. A value of zero indicates success. Other values are
1
maximum number of function evaluations exceeded
2
NPT, the number of approximation points, is not in the required interval
3
a trust region step failed to reduce q (Consult Powell for explanation.)
4
one of the box constraint ranges is too small (< 2*RHOBEG)
5
bobyqa detected too much cancellation in denominator (We have not fully understood Powell's code to explain this.)
msg
A message describing the outcome of UOBYQA

##### References

M. J. D. Powell (2007) "Developments of NEWUOA for unconstrained minimization without derivatives", Cambridge University, Department of Applied Mathematics and Theoretical Physics, Numerical Analysis Group, Report NA2007/05, http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2007_05.pdf.

M. J. D. Powell (2009), "The BOBYQA algorithm for bound constrained optimization without derivatives", Report No. DAMTP 2009/NA06, Centre for Mathematical Sciences, University of Cambridge, UK. http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf. Description was taken from comments in the Fortran code of M. J. D. Powell on which minqa is based.

optim, nlminb
library(minqa) fr <- function(x) { ## Rosenbrock Banana function 100 * (x - x^2)^2 + (1 - x)^2 } (x1 <- bobyqa(c(1, 2), fr, lower = c(0, 0), upper = c(4, 4))) ## => optimum at c(1, 1) with fval = 0 str(x1) # see that the error code and msg are returned # check the error exits # too many iterations x1e<-bobyqa(c(1, 2), fr, lower = c(0, 0), upper = c(4, 4), control = list(maxfun=50)) str(x1e) # Throw an error because bounds too tight x1b<-bobyqa(c(4,4), fr, lower = c(0, 3.9999999), upper = c(4, 4)) str(x1b) # Throw an error because npt is too small -- does NOT work as of 2010-8-10 as # minqa.R seems to force a reset. x1n<-bobyqa(c(2,2), fr, lower = c(0, 0), upper = c(4, 4), control=list(npt=1)) str(x1n) # To add if we can find them -- examples of ierr = 3 and ierr = 5.