optimx: axsearch – R documentation

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axsearch

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. While we believe that it will work using the wrapper ufn from this package with the 'maximize=TRUE' setting, we believe it is much safer to write your own function that is to be minimized. That is minimize (-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, trace=0, ...)

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 value of the objective function at the parameters `par`. ?? what if fmin==NULL?
`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).
`trace`	If trace>0, then local output is enabled.
`...`	Extra arguments for the user function.

Details

The axial search MAY give a lower function value, in which case, one can restart. 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.

Value

A list with components:

`bestfn`	The lowest (best) function value found (??maximize??) 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, 
                     trace=0)
print(axabtrvm)

abtrvm1 <- list() # set up structure
# Following can be executed if package optimx available
# cat("Now force an early stop\n")
# abtrvm1 <- optimr(xx, bt.f, bt.g, lower=lower, upper=upper, method="Rvmmin", 
#                   control=list(maxit=1, trace=0))
# print(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, 
                      trace=0)
print(axabtrvm1)

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

optimx

Expanded Replacement and Extension of the 'optim' Function

v2020-4.2

GPL-2

Authors

John C Nash [aut, cre], Ravi Varadhan [aut], Gabor Grothendieck [ctb]

Initial release

2020-04-02