Rvmmin - an R implementation of the Fletcher(1970)

In optimx: Expanded Replacement and Extension of the 'optim' Function

Rvmmin description, examples and tests

Rvmmin is an all-R version of the Fletcher-Nash variable metric nonlinear parameter optimization code of @Fletcher70 as modified by @cnm79.

This vignette is intended to show various features of the package, so it is rather detailed and "busy". However, it is also hopefully helpful in showing how to use the method for more difficult problems.

Algorithm implementation

Fletcher's variable metric method attempts to mimic Newton's iteration for function minimization approximately.

Newton's method starts with an original set of parameters $x_0$. At a given iteraion, which could be the first, we want to solve

$$x_{k+1} = x_{k} - H^{-1} g$$

where $H$ is the Hessian and $g$ is the gradient at $x_k$.

Newton's method is unattractive in general function minimization situations because

• evaluating the Hessian is generally time consuming and error prone;

• solving the equation $$H delta = -g$$ (which is much less computational effort than inverting $H$), is still a lot of work which needs to be carried out every iteration.

While the base Newton algorithm is as given, generally we carry out some sort of line search along the search direction delta from the current iterate $x_k$. Indeed, many otherwise highly educated workers try to implement it without paying attention to safeguarding the iterations and ensuring appropriate progress towards a minimum.

Termination nuances

Termination variation with control tolerances

Let us use the Chebyquad test problem in $n$=4 parameters with different controls eps and acctol and tabulate the results to explore how our results change with different values of these program control inputs.

cyq.f <- function (x) {
  rv<-cyq.res(x)
  f<-sum(rv*rv)
}

cyq.res <- function (x) {
# Fletcher's chebyquad function m = n -- residuals 
   n<-length(x)
   res<-rep(0,n) # initialize
   for (i in 1:n) { #loop over resids
     rr<-0.0
     for (k in 1:n) {
  z7<-1.0
  z2<-2.0*x[k]-1.0
        z8<-z2
        j<-1
        while (j<i) {
            z6<-z7
            z7<-z8
            z8<-2*z2*z7-z6 # recurrence to compute Chebyshev polynomial
            j<-j+1
        } # end recurrence loop
        rr<-rr+z8
      } # end loop on k
      rr<-rr/n
      if (2*trunc(i/2) == i) { rr <- rr + 1.0/(i*i - 1) }
      res[i]<-rr
    } # end loop on i
    res
}

cyq.jac<- function (x) {
#  Chebyquad Jacobian matrix
   n<-length(x)
   cj<-matrix(0.0, n, n)
   for (i in 1:n) { # loop over rows
     for (k in 1:n) { # loop over columns (parameters)
       z5<-0.0
       cj[i,k]<-2.0
       z8<-2.0*x[k]-1.0 
       z2<-z8
       z7<-1.0
       j<- 1
       while (j<i) { # recurrence loop
         z4<-z5
         z5<-cj[i,k]
         cj[i,k]<-4.0*z8+2.0*z2*z5-z4
         z6<-z7
         z7<-z8
         z8<-2.0*z2*z7-z6
         j<- j+1
       } # end recurrence loop
       cj[i,k]<-cj[i,k]/n
     } # end loop on k
   } # end loop on i
   cj
}


cyq.g <- function (x) {
   cj<-cyq.jac(x)
   rv<-cyq.res(x)
   gg<- as.vector(2.0* rv %*% cj)
}

require(optimx)
nn <- 4
xx0 <- 1:nn
xx0 <- xx0 / (nn+1.0) # Initial value suggested by Fletcher

# cat("aed\n")
# aed <- Rvmminu(xx0, cyq.f, cyq.g, control=list(trace=2, checkgrad=FALSE))
# print(aed)
#================================
# Now build a table of results for different values of eps and acc
veps <- c(1e-3, 1e-5, 1e-7, 1e-9, 1e-11)
vacc <- c(.1, .01, .001, .0001, .00001, .000001)
resdf <- data.frame(eps=NA, acctol=NA, nf=NA, ng=NA, fval=NA, gnorm=NA)
for (eps in veps) {
  for (acctol in vacc) {
    ans <- Rvmminu(xx0, cyq.f, cyq.g, 
          control=list(eps=eps, acctol=acctol, trace=0))
    gn <- as.numeric(crossprod(cyq.g(ans$par)))
    resdf <- rbind(resdf, 
              c(eps, acctol, ans$counts[1], ans$counts[2], ans$value, gn))
  }
}
resdf <- resdf[-1,]
# Display the function value found for different tolerances
xtabs(formula = fval ~ acctol + eps, data=resdf)
# Display the gradient norm found for different tolerances
xtabs(formula = gnorm ~ acctol + eps, data=resdf)
# Display the number of function evaluations used for different tolerances
xtabs(formula = nf ~ acctol + eps, data=resdf)
# Display the number of gradient evaluations used for different tolerances
xtabs(formula = ng ~ acctol + eps, data=resdf)

Here -- and we caution that this is but a single instance of a single test problem -- the differences in results and level of effort to obtain them are regulated by the values of eps only. This control is used to judge the size of the gradient norm and the gradient projection on the search vector.

Problems of function scale

One of the more difficult aspects of termination decisions is that we need to decide when we have a "nearly" zero gradient. However, this "zero gradient" is relative to the overall scale of the function. Let us see what happens when we consider solving a problem where the function scale is adjustable. Note that we multiply the constant sequence yy by pi/4 to avoid integer values which may give results that are fortuitously better than may be normally found.

sq<-function(x, exfs=1){
  nn<-length(x)
  yy<-(1:nn)*pi/4
  f<-(10^exfs)*sum((yy-x)^2)
  f
}
sq.g <- function(x, exfs=1){
  nn<-length(x)
  yy<-(1:nn)*pi/4
  gg<- 2*(x - yy)*(10^exfs)
}
require(optimx)
nn <- 4
xx0 <- rep(pi, nn) # crude start

# Now build a table of results for different values of eps and acc
veps <- c(1e-3, 1e-5, 1e-7, 1e-9, 1e-11)
exfsi <- 1:6
resdf <- data.frame(eps=NA, exfs=NA, nf=NA, ng=NA, fval=NA, gnorm=NA)
for (eps in veps) {
  for (exfs in exfsi) {
    ans <- Rvmminu(xx0, sq, sq.g, 
                   control=list(eps=eps, trace=0), exfs=exfs)
    gn <- as.numeric(crossprod(sq.g(ans$par)))
    resdf <- rbind(resdf, 
                   c(eps, exfs, ans$counts[1], ans$counts[2], ans$value, gn))
  }
}
resdf <- resdf[-1,]
# Display the function value found for different tolerances
xtabs(formula = fval ~ exfs + eps, data=resdf)
# Display the gradient norm found for different tolerances
xtabs(formula = gnorm ~ exfs + eps, data=resdf)
# Display the number of function evaluations used for different tolerances
xtabs(formula = nf ~ exfs + eps, data=resdf)
# Display the number of gradient evaluations used for different tolerances
xtabs(formula = ng ~ exfs + eps, data=resdf)

The general tendency here is for the amount of work in terms of function evaluations to rise with the function scale and with tighter (smaller) test tolerances, while the quality of the solution is poorer with larger scale and also larger (looser) tolerances. However, some exceptions can be seen, though the overall quality of solutions (function and gradient norm) is very good. Moreover, the number of gradient evaluations does not climb notably with the scale or inverse tolerance.

Problems of parameter scale

There are similar issues of parameter scaling. Let us look at very simple sum of squares function where we scale the parameters in a nasty way.

ssq.f<-function(x){
   nn<-length(x)
   yy <- 1:nn
   f<-sum((yy-x/10^yy)^2)
   f
}
ssq.g <- function(x){
   nn<-length(x)
   yy<-1:nn
   gg<- 2*(x/10^yy - yy)*(1/10^yy)
}

xy <- c(1, 1/10, 1/100, 1/1000)
# note: gradient was checked using numDeriv
veps <- c(1e-3, 1e-5, 1e-7, 1e-9, 1e-11)
vacc <- c(.1, .01, .001, .0001, .00001, .000001)
resdf <- data.frame(eps=NA, acctol=NA, nf=NA, ng=NA, fval=NA, gnorm=NA)
for (eps in veps) {
  for (acctol in vacc) {
    ans <- Rvmminu(xy, ssq.f, ssq.g, 
          control=list(eps=eps, acctol=acctol, trace=0))
    gn <- as.numeric(crossprod(ssq.g(ans$par)))
    resdf <- rbind(resdf, 
              c(eps, acctol, ans$counts[1], ans$counts[2], ans$value, gn))
  }
}
resdf <- resdf[-1,]
# Display the function value found for different tolerances
xtabs(formula = fval ~ acctol + eps, data=resdf)
# Display the gradient norm found for different tolerances
xtabs(formula = gnorm ~ acctol + eps, data=resdf)
# Display the number of function evaluations used for different tolerances
xtabs(formula = nf ~ acctol + eps, data=resdf)
# Display the number of gradient evaluations used for different tolerances
xtabs(formula = ng ~ acctol + eps, data=resdf)

The results above suggest that parameter scaling is not much of a problem. Actually, these are the very best results I have found with any method for this problem, which is actually rather nasty. I suggest trying this problem on your favourite optimizer. Alternatively, use the package optimr and run the function opm() with method="ALL".

Weeds problem with random starts

This notorious problem (see @cnm79, page 120, @nlpor14, page 205, for details under the heading Hobbs Weeds problem) is small but generally difficult due to the possibility of bad scaling of both function and parameters and a near-singular Hessian in the original parameterization.

The Fletcher variable metric method can solve this problem quite well, though default termination settings should be overridden. It is important to ensure there are enough iterations to allow the method to "grind" at the problem. If one uses default settings for maxit in optim:BFGS, then the success rate drops to less than 2/3 of cases tried below.

Below we use 100 "random" starting points for both Rvmmin and the optim:BFGS minimizers (which should be, but are not quite, the same).

## hobbstarts.R -- starting points for Hobbs problem 
hobbs.f<- function(x){ # # Hobbs weeds problem -- function
    if (abs(12*x[3]) > 500) { # check computability
       fbad<-.Machine$double.xmax
       return(fbad)
    }
    res<-hobbs.res(x)
    f<-sum(res*res)
##    cat("fval =",f,"\n")
##    f
}
hobbs.res<-function(x){ # Hobbs weeds problem -- residual
# This variant uses looping
    if(length(x) != 3) stop("hobbs.res -- parameter vector n!=3")
    y<-c(5.308, 7.24, 9.638, 12.866, 17.069, 23.192, 31.443, 
         38.558, 50.156, 62.948, 75.995, 91.972)
    t<-1:12
    if(abs(12*x[3])>50) {
       res<-rep(Inf,12)
    } else {
       res<-x[1]/(1+x[2]*exp(-x[3]*t)) - y
    }
}

hobbs.jac<-function(x){ # Jacobian of Hobbs weeds problem
   jj<-matrix(0.0, 12, 3)
   t<-1:12
    yy<-exp(-x[3]*t)
    zz<-1.0/(1+x[2]*yy)
     jj[t,1] <- zz
     jj[t,2] <- -x[1]*zz*zz*yy
     jj[t,3] <- x[1]*zz*zz*yy*x[2]*t
   return(jj)
}

hobbs.g<-function(x){ # gradient of Hobbs weeds problem
    # NOT EFFICIENT TO CALL AGAIN
    jj<-hobbs.jac(x)
    res<-hobbs.res(x)
    gg<-as.vector(2.*t(jj) %*% res)
    return(gg)
}
require(optimx)
set.seed(12345)
nrun<-100
sstart<-matrix(runif(3*nrun, 0, 5), nrow=nrun, ncol=3)
ustart<-sstart %*% diag(c(100, 10, 0.1))
nsuccR <- 0
nsuccO <- 0
vRvm <- rep(NA, nrun)
voptim <- vRvm
fRvm <- vRvm
gRvm <- vRvm
foptim <- vRvm
goptim <- vRvm

for (irun in 1:nrun) {
  us <- ustart[irun,]
#  print(us)
#  ans <- Rvmminu(us, hobbs.f, hobbs.g, control=list(trace=1))
#  ans <- optim(us, hobbs.f, hobbs.g, method="BFGS")
  ans <- Rvmminu(us, hobbs.f, hobbs.g, control=list(trace=0))
  ao <- optim(us, hobbs.f, hobbs.g, method="BFGS", 
               control=list(maxit=3000))
# ensure does not max function out

# cat(irun,"  Rvmminu value =",ans$value,"  optim:BFGS value=",ao$value,"\n")
  if (ans$value < 2.5879) nsuccR <- nsuccR + 1
  if (ao$value < 2.5879) nsuccO <- nsuccO + 1
#  tmp <- readline()
  vRvm[irun] <- ans$value
  voptim[irun] <- ao$value
  fRvm[irun] <- ans$counts[1]
  gRvm[irun] <- ans$counts[2]
  foptim[irun] <- ao$counts[1]
  goptim[irun] <- ao$counts[2]

}
cat("Rvmminu: number of successes=",nsuccR,"  propn=",nsuccR/nrun,"\n")
cat("optim:BFGS no. of successes=",nsuccO,"  propn=",nsuccO/nrun,"\n")
fgc <- data.frame(fRvm, foptim, gRvm, goptim)
summary(fgc)

From this summary, it appears that Rvmmin, on average, uses fewer gradient and function evaluations to achieve the desired result.

For comparison, we now re-run the example with default settings for maxit in optim:BFGS.

nsuccR <- 0
nsuccO <- 0
for (irun in 1:nrun) {
  us <- ustart[irun,]
#  print(us)
#  ans <- Rvmminu(us, hobbs.f, hobbs.g, control=list(trace=1))
#  ans <- optim(us, hobbs.f, hobbs.g, method="BFGS")
  ans <- Rvmminu(us, hobbs.f, hobbs.g, control=list(trace=0))
  ao <- optim(us, hobbs.f, hobbs.g, method="BFGS")
# ensure does not max function out

# cat(irun,"  Rvmminu value =",ans$value,"  optim:BFGS value=",ao$value,"\n")
  if (ans$value < 2.5879) nsuccR <- nsuccR + 1
  if (ao$value < 2.5879) nsuccO <- nsuccO + 1
#  tmp <- readline()
  vRvm[irun] <- ans$value
  voptim[irun] <- ao$value
  fRvm[irun] <- ans$counts[1]
  gRvm[irun] <- ans$counts[2]
  foptim[irun] <- ao$counts[1]
  goptim[irun] <- ao$counts[2]

}
cat("Rvmminu: number of successes=",nsuccR,"  propn=",nsuccR/nrun,"\n")
cat("optim:BFGS no. of successes=",nsuccO,"  propn=",nsuccO/nrun,"\n")
fgc <- data.frame(fRvm, foptim, gRvm, goptim)
summary(fgc)

Bounds and masks

Let us make sure that Rvmminb is doing the right thing with bounds and masks. (This is actually a test in the package.)

Bounds

bt.f<-function(x){
 sum(x*x)
}

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

lower <- c(0, 1, 2, 3, 4)
upper <- c(2, 3, 4, 5, 6)
bdmsk <- rep(1,5)
xx <- rep(0,5) # out of bounds
ans <- Rvmmin(xx, bt.f, bt.g, lower=lower, upper=upper, bdmsk=bdmsk)
ans

Masks

Here we fix one or more paramters and minimize over the rest.

sq.f<-function(x){
   nn<-length(x)
   yy<-1:nn
   f<-sum((yy-x)^2)
   f
}
sq.g <- function(x){
   nn<-length(x)
   yy<-1:nn
   gg<- 2*(x - yy)
}

xx0 <- rep(pi,3)
bdmsk <- c(1, 0, 1) # Middle parameter fixed at pi
cat("Check final function value (pi-2)^2 = ", (pi-2)^2,"\n")

require(optimx)
ans <- Rvmmin(xx0, sq.f, sq.g, lower=-Inf, upper=Inf, bdmsk=bdmsk,
               control=list(trace=2))
ans
ansnog <- Rvmmin(xx0, sq.f, lower=-Inf, upper=Inf, bdmsk=bdmsk,
               control=list(trace=2))
ansnog

References

optimx documentation built on June 14, 2019, 3:01 p.m.