axsearch {optimx}

Perform axial search around a supposed MINIMUM and provide diagnostics

Description

Nonlinear optimization problems often terminate at points in the parameter space that are not satisfactory optima. This routine conducts an axial search, stepping forward and backward along each parameter and computing the objective function. This allows us to compute the tilt and radius of curvature or roc along that parameter axis.

axsearch assumes that one is MINIMIZING the function fn. If you are working with a maximization, it is suggested that you write your own function that is to be minimized, that is, (-1)*(function to be maximized). All discussion here is in terms of minimization.

Axial search may find parameters with a function value lower than that at the supposed minimum, i.e., lower than fmin.

In this case axsearch exits immediately with the new function value and parameters. This can be used to restart an optimizer, as in the optimx wrapper.

Usage

   axsearch(par, fn=NULL, fmin=NULL, lower=NULL, upper=NULL, bdmsk=NULL, 
              control=list(), ...)

Arguments

par

A numeric vector of values of the optimization function parameters that are at a supposed minimum.

fn

The user objective function

fmin

The presumed value of the objective function at the parameters par. NOTE: This is NOT cheched. Caution!. However, if fmin==NULL on call, axsearch will compute the value.

lower

A vector of lower bounds on the parameters.

upper

A vector of upper bounds on the parameters.

bdmsk

An indicator vector, having 1 for each parameter that is "free" or unconstrained, and 0 for any parameter that is fixed or MASKED for the duration of the optimization. Partly for historical reasons, we use the same array during the progress of optimization as an indicator that a parameter is at a lower bound (bdmsk element set to -3) or upper bound (-1).

control

Algorithm controls as per ctrldefault. See details.

...

Extra arguments for the user function.

Details

The axial search MAY give a lower function value, in which case, one can restart an optimization. However, it is left to the user to do this. Its primary use is in presenting some features of the function surface in the tilt and radius of curvature measures returned. However, better measures should be possible, and this function should be regarded as largely experimental.

Note: As of December 2021, the calling syntax has changed from axsearch(par, fn=NULL, fmin=NULL, lower=NULL, upper=NULL, bdmsk=NULL, trace=0, ...)

In case any user has code employing the older function, it is to be found in inst/doc/replaced2021/axsearch2018.R.

The new syntax has trace replaced with control=list{}, where the defaults are found from the function ctrldefault(). This routine uses three particular elements:

trace is 0 if no intermediate output is desired, non-zero otherwise.

bigval is a large number used to provide a value for the objective function when the parameters are inadmissible.

reltest is used to test for equality of small numbers by comparing their sums with reltest.

grtesttol is a small quantity, but it is used when multiplied by reltest to give epst, the axial step control. Each parameter is stepped by an amount epst*(abs(parameter_value)+epst). Note that the author has never found it necessary to adjust these values from the defaults generated by ctrldefault().

Value

A list with components:

bestfn

The lowest (best) function value found during the axial search, else the original fmin value. (This is actively set in that case.)

par

The vector of parameters at the best function value.

details

A data frame reporting the original parameters, the forward step and backward step function values, the size of the step taken for a particular parameter, the tilt and the roc (radius of curvature). Some elements will be NA if we find a lower function value during the axial search.

Examples

#####################
# require(optimx)
# Simple bounds test for n=4
bt.f<-function(x){
  sum(x*x)
}

bt.g<-function(x){
  gg<-2.0*x
}

n<-4
lower<-rep(0,n)
upper<-lower # to get arrays set
bdmsk<-rep(1,n)
# bdmsk[(trunc(n/2)+1)]<-0
for (i in 1:n) { 
  lower[i]<-1.0*(i-1)*(n-1)/n
  upper[i]<-1.0*i*(n+1)/n
}
xx<-0.5*(lower+upper)

cat("lower bounds:")
print(lower)
cat("start:       ")
print(xx)
cat("upper bounds:")
print(upper)

abtrvm <- list() # ensure we have the structure

cat("Rvmmin \n\n")
# Note: trace set to 0 below. Change as needed to view progress. 

# Following can be executed if package optimx available
# abtrvm <- optimr(xx, bt.f, bt.g, lower=lower, upper=upper, method="Rvmmin", 
#                 control=list(trace=0))
# Note: use lower=lower etc. because there is a missing hess= argument
# print(abtrvm)

abtrvm$par <- c(0.00, 0.75, 1.50, 2.25)
abtrvm$value <- 7.875
cat("Axial search")
axabtrvm <- axsearch(abtrvm$par, fn=bt.f, fmin=abtrvm$value, lower, upper, bdmsk=NULL)
print(axabtrvm)

abtrvm1 <- optimr(xx, bt.f, bt.g, lower=lower, upper=upper, method="Rvmmin", 
                   control=list(maxit=1, trace=0))
proptimr(abtrvm1)

abtrvm1$value <- 8.884958
abtrvm1$par <- c(0.625, 1.625, 2.625, 3.625)

cat("Axial search")
axabtrvm1 <- axsearch(abtrvm1$par, fn=bt.f, fmin=abtrvm1$value, lower, upper, bdmsk=NULL)
print(axabtrvm1)

cat("Do NOT try axsearch() with maximize\n")