Rvmmin is an all-R version of the Fletcher-Nash variable metric nonlinear parameter optimization code of Fletcher (1970) ?? ref needed as modified by Nash (1979) ??ref needed.
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.
We use Chebyquad n = 4 test with different controls eps and acctol and tabulate the results.
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(Rvmmin)
## Loading required package: Rvmmin
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)
#================================
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,]
# print(resdf)
xtabs(formula = fval ~ acctol + eps, data=resdf)
## eps
## acctol 1e-11 1e-09 1e-07 1e-05 0.001
## 1e-06 3.964816e-29 3.964816e-29 3.964816e-29 7.049696e-24 7.486504e-15
## 1e-05 3.964816e-29 3.964816e-29 3.964816e-29 7.049696e-24 7.486504e-15
## 1e-04 3.964816e-29 3.964816e-29 3.964816e-29 7.049696e-24 7.486504e-15
## 0.001 3.964816e-29 3.964816e-29 3.964816e-29 7.049696e-24 7.486504e-15
## 0.01 3.964816e-29 3.964816e-29 3.964816e-29 7.049696e-24 7.486504e-15
## 0.1 3.964816e-29 3.964816e-29 3.964816e-29 7.049696e-24 7.486504e-15
xtabs(formula = gnorm ~ acctol + eps, data=resdf)
## eps
## acctol 1e-11 1e-09 1e-07 1e-05 0.001
## 1e-06 7.809261e-30 7.809261e-30 7.809261e-30 3.645064e-22 1.089927e-13
## 1e-05 7.809261e-30 7.809261e-30 7.809261e-30 3.645064e-22 1.089927e-13
## 1e-04 7.809261e-30 7.809261e-30 7.809261e-30 3.645064e-22 1.089927e-13
## 0.001 7.809261e-30 7.809261e-30 7.809261e-30 3.645064e-22 1.089927e-13
## 0.01 7.809261e-30 7.809261e-30 7.809261e-30 3.645064e-22 1.089927e-13
## 0.1 7.809261e-30 7.809261e-30 7.809261e-30 3.645064e-22 1.089927e-13
xtabs(formula = nf ~ acctol + eps, data=resdf)
## eps
## acctol 1e-11 1e-09 1e-07 1e-05 0.001
## 1e-06 22 22 22 17 12
## 1e-05 22 22 22 17 12
## 1e-04 22 22 22 17 12
## 0.001 22 22 22 17 12
## 0.01 22 22 22 17 12
## 0.1 22 22 22 17 12
xtabs(formula = ng ~ acctol + eps, data=resdf)
## eps
## acctol 1e-11 1e-09 1e-07 1e-05 0.001
## 1e-06 15 15 15 12 9
## 1e-05 15 15 15 12 9
## 1e-04 15 15 15 12 9
## 0.001 15 15 15 12 9
## 0.01 15 15 15 12 9
## 0.1 15 15 15 12 9
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 and its parameters.
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: checked gradient 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,]
# print(resdf)
xtabs(formula = fval ~ acctol + eps, data=resdf)
## eps
## acctol 1e-11 1e-09 1e-07 1e-05 0.001
## 1e-06 0.000000e+00 0.000000e+00 1.475416e-29 5.767419e-19 8.977439e-11
## 1e-05 0.000000e+00 0.000000e+00 1.475416e-29 5.767419e-19 8.977439e-11
## 1e-04 0.000000e+00 0.000000e+00 1.475416e-29 5.767419e-19 8.977439e-11
## 0.001 0.000000e+00 0.000000e+00 1.475416e-29 5.767419e-19 8.977439e-11
## 0.01 0.000000e+00 0.000000e+00 1.475416e-29 5.767419e-19 8.977439e-11
## 0.1 0.000000e+00 0.000000e+00 1.475416e-29 5.767419e-19 8.977439e-11
xtabs(formula = gnorm ~ acctol + eps, data=resdf)
## eps
## acctol 1e-11 1e-09 1e-07 1e-05 0.001
## 1e-06 0.000000e+00 0.000000e+00 7.783028e-33 3.430257e-23 3.473135e-14
## 1e-05 0.000000e+00 0.000000e+00 7.783028e-33 3.430257e-23 3.473135e-14
## 1e-04 0.000000e+00 0.000000e+00 7.783028e-33 3.430257e-23 3.473135e-14
## 0.001 0.000000e+00 0.000000e+00 7.783028e-33 3.430257e-23 3.473135e-14
## 0.01 0.000000e+00 0.000000e+00 7.783028e-33 3.430257e-23 3.473135e-14
## 0.1 0.000000e+00 0.000000e+00 7.783028e-33 3.430257e-23 3.473135e-14
xtabs(formula = nf ~ acctol + eps, data=resdf)
## eps
## acctol 1e-11 1e-09 1e-07 1e-05 0.001
## 1e-06 56 56 55 53 51
## 1e-05 56 56 55 53 51
## 1e-04 56 56 55 53 51
## 0.001 56 56 55 53 51
## 0.01 56 56 55 53 51
## 0.1 56 56 55 53 51
xtabs(formula = ng ~ acctol + eps, data=resdf)
## eps
## acctol 1e-11 1e-09 1e-07 1e-05 0.001
## 1e-06 56 56 55 53 51
## 1e-05 56 56 55 53 51
## 1e-04 56 56 55 53 51
## 0.001 56 56 55 53 51
## 0.01 56 56 55 53 51
## 0.1 56 56 55 53 51
This notorious problem (see Nash, 1979 or Nash, 2014 for details under the Hobbs Weeds problem) is small but generally difficult due to bad scaling and a near-singular Hessian in the original parameterization.
However, the Fletcher variable metric method can solve this problem quite well. It is important to ensure there are enough iterations to allow the method to “grind” at the problem. If one uses default settings for various aspects of the termination criteria (maxit in optim:BFGS or stopbadupdate=TRUE in Rvmminu), then the success rate drops to less than 2/3 of cases tried below.
## 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$double.xmax
return(fbad)
}
res<-hobbs.res(x)
f<-sum(res*res)
## cat("fval =",f,"\n")
## f
}
hobbs.res<-function(x){ # Hobbs weeds problem -- residual
# This variant uses looping
if(length(x) != 3) stop("hobbs.res -- parameter vector n!=3")
y<-c(5.308, 7.24, 9.638, 12.866, 17.069, 23.192, 31.443,
38.558, 50.156, 62.948, 75.995, 91.972)
t<-1:12
if(abs(12*x[3])>50) {
res<-rep(Inf,12)
} else {
res<-x[1]/(1+x[2]*exp(-x[3]*t)) - y
}
}
hobbs.jac<-function(x){ # Jacobian of Hobbs weeds problem
jj<-matrix(0.0, 12, 3)
t<-1:12
yy<-exp(-x[3]*t)
zz<-1.0/(1+x[2]*yy)
jj[t,1] <- zz
jj[t,2] <- -x[1]*zz*zz*yy
jj[t,3] <- x[1]*zz*zz*yy*x[2]*t
return(jj)
}
hobbs.g<-function(x){ # gradient of Hobbs weeds problem
# NOT EFFICIENT TO CALL AGAIN
jj<-hobbs.jac(x)
res<-hobbs.res(x)
gg<-as.vector(2.*t(jj) %*% res)
return(gg)
}
require(Rvmmin)
set.seed(12345)
nrun<-100
sstart<-matrix(runif(3*nrun, 0, 5), nrow=nrun, ncol=3)
ustart<-sstart %*% diag(c(100, 10, 0.1))
nsuccR <- 0
nsuccO <- 0
vR <- rep(NA, nrun)
vO <- vR
fR <- vR
gR <- vR
fO <- vR
gO <- vR
for (irun in 1:nrun) {
us <- ustart[irun,]
print(us)
# ans <- Rvmminu(us, hobbs.f, hobbs.g, control=list(trace=1))
# ans <- optim(us, hobbs.f, hobbs.g, method="BFGS")
ans <- Rvmminu(us, hobbs.f, hobbs.g, control=list(trace=0))
ao <- optim(us, hobbs.f, hobbs.g, method="BFGS",
control=list(maxit=3000))
# ensure does not max function out
cat(irun," Rvmminu value =",ans$value," optim:BFGS value=",ao$value,"\n")
if (ans$value < 2.5879) nsuccR <- nsuccR + 1
if (ao$value < 2.5879) nsuccO <- nsuccO + 1
# tmp <- readline()
vR[irun] <- ans$value
vO[irun] <- ao$value
fR[irun] <- ans$counts[1]
gR[irun] <- ans$counts[2]
fO[irun] <- ao$counts[1]
gO[irun] <- ao$counts[2]
}
## [1] 360.4519481 14.7232702 0.2942962
## 1 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 437.8865965 30.8626833 0.4462959
## 2 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 380.49116416 48.71370636 0.06189745
## 3 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 443.0622831 30.9106019 0.2566545
## 4 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 228.2404801 26.0684597 0.3318201
## 5 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 83.185893 45.125190 0.382771
## 6 Rvmminu value = 2.587277 optim:BFGS value= 2.58728
## [1] 162.54769336 31.87272163 0.04631309
## 7 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 254.61216783 43.21505665 0.03382227
## 8 Rvmminu value = 2.587277 optim:BFGS value= 2.587279
## [1] 363.8526269 12.5558870 0.2780492
## 9 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 494.8684690 10.7534540 0.3256592
## 10 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 17.2677175 30.4738022 0.1479813
## 11 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 76.1867451 19.1722318 0.3477524
## 12 Rvmminu value = 2.587277 optim:BFGS value= 2.587281
## [1] 367.842476 37.763552 0.188835
## 13 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 0.5682933 18.9868121 0.1304810
## 14 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 195.6016676 39.7486032 0.4700795
## 15 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 231.247327 45.284557 0.215445
## 16 Rvmminu value = 2.587277 optim:BFGS value= 2.58729
## [1] 194.0719908 49.2013087 0.1635323
## 17 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 201.2425709 29.3973987 0.2674541
## 18 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 89.4817924 0.4732026 0.3129708
## 19 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 475.82937684 16.03960346 0.08521099
## 20 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 226.86403664 27.97283666 0.05465815
## 21 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 163.37620432 25.74587801 0.05363939
## 22 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 482.70766169 4.48559311 0.04637089
## 23 Rvmminu value = 2.587277 optim:BFGS value= 2.587279
## [1] 353.7409386 34.1625287 0.2672878
## 24 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 322.2713183 35.4986025 0.4481946
## 25 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 194.914242 40.011578 0.355013
## 26 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 349.2718197 47.2039207 0.4532271
## 27 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 272.0289322 9.3323069 0.2183443
## Warning in Rvmminu(us, hobbs.f, hobbs.g, control = list(trace = 0)): Too
## many gradient evaluations
## 28 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 113.2335893 13.9321141 0.2339094
## 29 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 242.2788775 38.7131644 0.2990598
## 30 Rvmminu value = 2.587277 optim:BFGS value= 2.58728
## [1] 396.5035849 1.6723807 0.1533739
## 31 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 2.9938148 33.5959070 0.2860004
## 32 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 93.8562229 40.6383561 0.2396016
## 33 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 340.9168116 33.1580619 0.3483085
## 34 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 185.0520618 47.2237336 0.3676399
## 35 Rvmminu value = 2.587277 optim:BFGS value= 2.587279
## [1] 180.8127851 43.7422034 0.2582818
## 36 Rvmminu value = 2.587277 optim:BFGS value= 2.58728
## [1] 434.3974493 34.8685599 0.4926857
## 37 Rvmminu value = 2.587277 optim:BFGS value= 2.587279
## [1] 452.0773329 9.2520446 0.4589896
## 38 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 308.712283 4.877216 0.236244
## 39 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 67.0158173 40.3879995 0.1903133
## 40 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 391.0966403 26.0888818 0.4694052
## 41 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 214.5994100 34.6770338 0.4722385
## 42 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 463.6369875 10.8164795 0.4534123
## 43 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 386.6216124 31.4971436 0.3921888
## 44 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 129.8406234 7.3503939 0.3131655
## 45 Rvmminu value = 2.587277 optim:BFGS value= 2.587279
## [1] 160.6123367 47.9535230 0.3378219
## 46 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 30.0975787 49.5055223 0.4685549
## 47 Rvmminu value = 2.587277 optim:BFGS value= 2.587321
## [1] 21.7282269 24.8219654 0.2672636
## 48 Rvmminu value = 2.587277 optim:BFGS value= 2.587279
## [1] 27.526909 45.982382 0.168395
## 49 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 312.7713985 18.6065070 0.3953354
## 50 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 482.23514436 41.35287229 0.06163229
## 51 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 413.6514345 27.9521474 0.4795642
## 52 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 157.514118 39.783560 0.016246
## 53 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 106.5127255 17.6010873 0.1298446
## 54 Rvmminu value = 2.587277 optim:BFGS value= 2.587311
## [1] 366.2480594 10.4263836 0.4382151
## 55 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 249.6205103 39.2390312 0.4228122
## 56 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 3.648860e+02 3.415256e+01 8.426952e-03
## 57 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 40.16802215 21.60055070 0.09182044
## 58 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 217.76524244 42.32169935 0.05792107
## 59 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 118.2902267 30.0971920 0.2392073
## 60 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 395.7838992 37.0330258 0.4748624
## 61 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 129.3421584 29.7451001 0.4128688
## 62 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 492.9919159 42.5779545 0.4057794
## 63 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 378.4368718 40.6950504 0.4578685
## 64 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 489.8891236 19.0211676 0.3500826
## 65 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 109.47391961 24.61541012 0.04151929
## 66 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 474.35359156 49.66887323 0.05903807
## 67 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 74.7289731 8.7044725 0.3968248
## 68 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 300.1784853 41.7140400 0.4980219
## 69 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 473.2153759 19.7907687 0.3455178
## 70 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 344.17667845 48.45280671 0.07366959
## 71 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 252.7668616 9.4103233 0.1589239
## 72 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 186.46812241 37.46108349 0.07798945
## 73 Rvmminu value = 2.587277 optim:BFGS value= 2.58728
## [1] 167.9025192 15.4234805 0.4976268
## 74 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 24.1256763 18.9529451 0.3741136
## 75 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 309.4737695 37.5382922 0.1965197
## 76 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 480.7236459 35.2100838 0.2756957
## 77 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 327.48025481 46.02965168 0.01470503
## 78 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 255.145996 39.763866 0.168151
## 79 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 75.0491053 41.4538482 0.4291896
## 80 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 435.22394018 8.37738313 0.06354407
## 81 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 257.2208389 20.4979890 0.3957472
## 82 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 4.32395679 9.81263703 0.04894095
## 83 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 9.597382741 12.970031216 0.009576733
## 84 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 72.2559508 48.5208133 0.1380112
## 85 Rvmminu value = 2.587277 optim:BFGS value= 2.587411
## [1] 152.515877 40.582160 0.205576
## 86 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 412.82843135 17.37979198 0.03335091
## 87 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 251.1723212 46.9172764 0.3982658
## 88 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 401.7863129 8.1693447 0.2531626
## 89 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 30.3199911 7.8404580 0.2593119
## 90 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 463.97757 37.70316 0.35273
## 91 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 404.0892755 44.3025201 0.4158095
## 92 Rvmminu value = 2.587277 optim:BFGS value= 18.45938
## [1] 39.40666991 15.33820682 0.03044822
## 93 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 300.4641408 15.5540874 0.1684179
## 94 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 357.3899403 49.2262595 0.4011205
## 95 Rvmminu value = 2.587277 optim:BFGS value= 2.587281
## [1] 256.9228266 0.3464615 0.1417513
## 96 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 360.0582525 13.7160289 0.4074386
## 97 Rvmminu value = 2.587277 optim:BFGS value= 2.587277
## [1] 374.9731350 39.9022107 0.1075416
## 98 Rvmminu value = 2.587277 optim:BFGS value= 2.587278
## [1] 47.8202619 12.1834859 0.2775764
## 99 Rvmminu value = 2.587277 optim:BFGS value= 2.587316
## [1] 1.989129e+02 2.291367e+01 7.763851e-03
## 100 Rvmminu value = 2.587277 optim:BFGS value= 2.587279
cat("Rvmminu: number of successes=",nsuccR," propn=",nsuccR/nrun,"\n")
## Rvmminu: number of successes= 100 propn= 1
cat("optim:BFGS no. of successes=",nsuccO," propn=",nsuccO/nrun,"\n")
## optim:BFGS no. of successes= 99 propn= 0.99
fgc <- data.frame(fR, fO, gR, gO)
# print(fgc)
summary(fgc)
## fR fO gR gO
## Min. : 41.0 Min. : 58.0 Min. : 26.00 Min. : 16.0
## 1st Qu.:105.8 1st Qu.: 140.5 1st Qu.: 39.00 1st Qu.: 53.0
## Median :155.5 Median : 184.0 Median : 53.00 Median : 68.5
## Mean :205.7 Mean : 323.5 Mean : 59.57 Mean :131.2
## 3rd Qu.:258.0 3rd Qu.: 453.5 3rd Qu.: 66.00 3rd Qu.:178.8
## Max. :920.0 Max. :1427.0 Max. :507.00 Max. :610.0
Let us make sure that Rvmminb is doing the right thing with bounds and masks.
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)
## Warning in Rvmmin(xx, bt.f, bt.g, lower = lower, upper = upper, bdmsk =
## bdmsk): Parameter out of bounds has been moved to nearest bound
ans
## $par
## [1] 0 1 2 3 4
##
## $value
## [1] 30
##
## $counts
## function gradient
## 1 1
##
## $convergence
## [1] 0
##
## $message
## [1] "Rvmminb appears to have converged"
##
## $bdmsk
## [1] 1 -3 -3 -3 -3
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")
## Check final function value (pi-2)^2 = 1.303234
require(Rvmmin)
ans <- Rvmmin(xx0, sq.f, sq.g, lower=-Inf, upper=Inf, bdmsk=bdmsk,
control=list(trace=2))
## Bounds: nolower = TRUE noupper = TRUE bounds = TRUE
## Gradient test with tolerance = 6.055454e-06
## Analytic gradient uses function gr
## function at parameters = 5.909701 with attributes:
## NULL
## Compute analytic gradient
## [1] 4.2831853 2.2831853 0.2831853
## Compute numeric gradient
## [1] 4.2831853 2.2831853 0.2831853
## gradient test tolerance = 6.055454e-06 fval= 5.909701
## compare to max(abs(gn-ga))/(1+abs(fval)) = 3.242827e-12
## admissible = TRUE
## maskadded = FALSE
## parchanged = FALSE
## Bounds: nolower = FALSE noupper = FALSE bounds = TRUE
## Rvmminb -- J C Nash 2009-2015 - an R implementation of Alg 21
## Problem of size n= 3 Dot arguments:
## list()
## Initial fn= 5.909701
## 1 1 5.909701
## Gradproj = -18.42587
## reset steplength= 1
## *reset steplength= 0.2
## ig= 2 gnorm= 2.575522 3 2 2.961562
## Gradproj = -15.04576
## reset steplength= 1
## *reset steplength= 0.2
## ig= 3 gnorm= 0.23879 5 3 1.317489
## Gradproj = -0.02851034
## reset steplength= 1
## ig= 4 gnorm= 0 Small gradient norm
## Seem to be done Rvmminb
ans
## $par
## [1] 1.000000 3.141593 3.000000
##
## $value
## [1] 1.303234
##
## $counts
## function gradient
## 6 4
##
## $convergence
## [1] 2
##
## $message
## [1] "Rvmminb appears to have converged"
##
## $bdmsk
## [1] 1 0 1
ansnog <- Rvmmin(xx0, sq.f, lower=-Inf, upper=Inf, bdmsk=bdmsk,
control=list(trace=2))
## Bounds: nolower = TRUE noupper = TRUE bounds = TRUE
## WARNING: forward gradient approximation being used
## admissible = TRUE
## maskadded = FALSE
## parchanged = FALSE
## Bounds: nolower = FALSE noupper = FALSE bounds = TRUE
## Rvmminb -- J C Nash 2009-2015 - an R implementation of Alg 21
## Problem of size n= 3 Dot arguments:
## list()
## WARNING: using gradient approximation ' grfwd '
## Initial fn= 5.909701
## 1 1 5.909701
## Gradproj = -18.42587
## reset steplength= 1
## *reset steplength= 0.2
## ig= 2 gnorm= 2.575522 3 2 2.961562
## Gradproj = -15.04576
## reset steplength= 1
## *reset steplength= 0.2
## ig= 3 gnorm= 0.23879 5 3 1.317489
## Gradproj = -0.02851034
## reset steplength= 1
## ig= 4 gnorm= 2.668644e-08 6 4 1.303234
## Gradproj = -4.446061e-16
## reset steplength= 1
## *reset steplength= 0.2
## *reset steplength= 0.04
## *reset steplength= 0.008
## *reset steplength= 0.0016
## *reset steplength= 0.00032
## *reset steplength= 6.4e-05
## *reset steplength= 1.28e-05
## *reset steplength= 2.56e-06
## *reset steplength= 5.12e-07
## *reset steplength= 1.024e-07
## Unchanged in step redn
## No acceptable point
## Reset to gradient search
## 16 4 1.303234
## Gradproj = -7.121661e-16
## reset steplength= 1
## *reset steplength= 0.2
## *reset steplength= 0.04
## *reset steplength= 0.008
## *reset steplength= 0.0016
## *reset steplength= 0.00032
## *reset steplength= 6.4e-05
## *reset steplength= 1.28e-05
## *reset steplength= 2.56e-06
## *reset steplength= 5.12e-07
## *reset steplength= 1.024e-07
## Unchanged in step redn
## No acceptable point
## Converged
## Seem to be done Rvmminb
ansnog
## $par
## [1] 1.000000 3.141593 3.000000
##
## $value
## [1] 1.303234
##
## $counts
## function gradient
## 26 4
##
## $convergence
## [1] 0
##
## $message
## [1] "Rvmminb appears to have converged"
##
## $bdmsk
## [1] 1 0 1