So-called Newton methods are among the most commonly mentioned in the solution of nonlinear equations or function minimization. However, as discussed in
https://en.wikipedia.org/wiki/Newton%27s_method#History,
the Newton or Newton-Raphson method as we know it today was not what either of its supposed originators knew.
This vignette discusses the development of simple safeguarded variants of the Newton method for function minimization in R. Note that there are some resources in R for solving nonlinear equations by Newton-like methods in the packages nleqslv and pracma.
If we have a function \(f(x)\), with gradient \(g(x)\) and second derivative (Hessian) \(H(x)\) the first order condition for an extremum (min or max) is
\[g(x) = 0\]
To ensure a minimum, we want
\[ H(x) > 0 \]
The first order condition leads to a root-finding problem.
It turns out that \(x\) need not be a scalar. We can consider it to be a vector of parameters to be determined. This renders \(g(x)\) a vector also, and \(H(x)\) a matrix. The conditions of optimality then require a zero gradient and positive-definite Hessian.
The Newton approach to such equations is to provide a guess to the root \(x_try\) and to then solve the equation
\[ H(x_t) * s = - g(x_t)\]
for the search vector \(s\). We update \(x_t\) to \(x_t + s\) and repeat until we have a very small gradient \(g(x_t)\). If \(H(x)\) is positive definite, we have a reasonable approximation to a (local) minimum.
A particular interest in Newton-like methods its theoretical quadratic convergence. See https://en.wikipedia.org/wiki/Newton%27s_method. That is, the method will converge in one step for a quadratic function \(f(x)\), and for “reasonable” functions will converge very rapidly. There are, however, a number of conditions, and practical programs need to include safequards against mis-steps in the iterations.
The principal issues concern the possiblity that \(H(x)\) may not be positive definite, at least in some parts of the domain, and that the curvature may be such that a unit step \(x_t + s\) does not reduce the function \(f\). We therefore get a number of possible variants of the method when different possible safeguards are applied.
There are many choices we can make in building a practical code to implement the ideas above. In tandem with the two main issues expressed above, we will consider
the modification of the solution of the main equation \[ H(x_t) * s = - g(x_t)\] so that a reasonable search vector \(s\) is always generated by avoiding Hessian matrices that are not positive definite.
the selection of a new set of parameters \(x_new = x_t + step * s\) so that the function value \(f(x_new)\) is less than \(f(x_t)\).
The second choice above could be made slightly more stringent so that the Armijo condition of sufficient-decrease is met. Adding a curvature requirement gives the Wolfe condisions. See https://en.wikipedia.org/wiki/Wolfe_conditions. The Armijo requirement is generally written
\[f(x_t + step*s) < f(x_t) + c * step * g(x_t)^T*s\]
where c is some number less than 1. Typically \(c = 1e-4 = 0.0001\). Note that the product of gradient times search vector is negative for any reasonable situation, since we are trying to go “downhill”.
As a result of the ideas in this section, the code snewton() uses a solution of
the Newton equations with the Hessian provided (if this is possible, else we stop),
along with a backtracking line search. The code snewtonm uses a Marquardt
stabilization of the Hessian to create
\[ Haug = H + 1_n * lambda\]
That is, we add \(lambda\) times the unit matrix to \(H\). Then we try the set of parameters
found by adding the solution of the Newton equations with \(Haug\) in place of \(H\) to the
current “best” set of parameters. If this new set of parameters has a higher function
value than the “best” so far, we increase \(lambda\) and try again. Note that we do not
need to re-evaluate the gradient or Hessian to do this. Moreover, for some value of
\(lambda\), the step is clearly down the gradient (i.e., steepest descents) or we have
converged and no progress is possible. This leads to a very compact and elegant
code, which we name snewtonm() for Safeguarded Newton-Marquardt method.
It is reliable, but may be less efficient than using the un-modified Hessian.
The primary concern in solving for \(s\) is that the Hessian may not be positive definite. This means that we cannot apply fast and stable methods like the Cholesky decomposition to the matrix. At the time of writing, we use the following approach:
We attempt to solve
\[ H(x_t) * s = - g(x_t)\]
with R directly, and rely on internal
checks to catch any cases where the solution fails. We then use try() to stop the
program in this case.
The traditional Newton approach is that the stepsize is taken to be 1. In practice,
this can sometimes mean that the function value is not reduced. As an alternative,
we can use a simple backtrack search. We start with \(step = 1\) (actually the program
allows for the element defstep of the control list to be set to a value other
than 1). If the Armijo condition is not met, we replace \(step\) with $ r * step $
where \(r\) is less than 1. Here we suggest control$stepdec = 0.2. We repeat until
\(x_t\) satisfies the Armijo condition or \(x_t\) is essentially unchanged by the step.
Here “essentially unchanged” is determined by a test using an offset value, that is, the test
\[ (x_t + offset) == (x_t + step * d + offset) \]
where \(d\) is the search direction. control$offset = 100 is used. We could
also, and almost equivalently, use the R identical function.
This approach has been coded into the snewton() function.
These examples are coded as a test to the interim package snewton, but as
at 2018-7-10 are part of the optimx package. We call these below mostly via
the optimr() function to allow compact output to be used, but please note that
some count information on the number of hessian evaluations and “iterations” (which
generally is an algorithm-specific measure) is not then returned.
The following example is trivial, in that the Hessian is a constant matrix, and we achieve convergence immediately.
x0<-c(1,2,3,4)
fnt <- function(x, fscale=10){
yy <- length(x):1
val <- sum((yy*x)^2)*fscale
}
grt <- function(x, fscale=10){
nn <- length(x)
yy <- nn:1
# gg <- rep(NA,nn)
gg <- 2*(yy^2)*x*fscale
gg
}
hesst <- function(x, fscale=10){
nn <- length(x)
yy <- nn:1
hh <- diag(2*yy^2*fscale)
hh
}
require(optimx)
sessionInfo() # included to ensure we know which version of optimx in use
## R version 3.6.3 (2020-02-29)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Linux Mint 19.3
##
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.7.1
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.7.1
##
## locale:
## [1] LC_CTYPE=en_CA.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_CA.UTF-8 LC_COLLATE=C
## [5] LC_MONETARY=en_CA.UTF-8 LC_MESSAGES=en_CA.UTF-8
## [7] LC_PAPER=en_CA.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_CA.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] optimx_2020-4.2
##
## loaded via a namespace (and not attached):
## [1] Rcpp_1.0.4 digest_0.6.25 magrittr_1.5
## [4] evaluate_0.14 rlang_0.4.5 stringi_1.4.6
## [7] rmarkdown_2.1 tools_3.6.3 stringr_1.4.0
## [10] tinytex_0.20 markdown_1.1 numDeriv_2016.8-1.1
## [13] xfun_0.12 yaml_2.2.1 compiler_3.6.3
## [16] htmltools_0.4.0 knitr_1.28
t1 <- snewton(x0, fnt, grt, hesst, control=list(trace=0), fscale=3.0)
print(t1)
## $par
## [1] 0 0 0 0
##
## $value
## [1] 0
##
## $grad
## [1] 0 0 0 0
##
## $Hess
## [,1] [,2] [,3] [,4]
## [1,] 96 0 0 0
## [2,] 0 54 0 0
## [3,] 0 0 24 0
## [4,] 0 0 0 6
##
## $counts
## $counts$niter
## [1] 2
##
## $counts$nfn
## [1] 2
##
## $counts$ngr
## [1] 2
##
## $counts$nhess
## [1] 1
##
##
## $convcode
## [1] 0
##
## $message
## [1] "Small gradient norm"
# we can also use nlm and nlminb
fght <- function(x, fscale=10){
## combine f, g and h into single function for nlm
ff <- fnt(x, fscale)
gg <- grt(x, fscale)
hh <- hesst(x, fscale)
attr(ff, "gradient") <- gg
attr(ff, "hessian") <- hh
ff
}
t1nlm <- nlm(fght, x0, fscale=3.0, hessian=TRUE, print.level=0)
print(t1nlm)
## $minimum
## [1] 1.232595e-29
##
## $estimate
## [1] 0.000000e+00 2.220446e-16 4.440892e-16 0.000000e+00
##
## $gradient
## [1] 0.000000e+00 3.996803e-14 3.552714e-14 0.000000e+00
##
## $hessian
## [,1] [,2] [,3] [,4]
## [1,] 3.200000e+02 0 0 4.235165e-14
## [2,] 0.000000e+00 180 0 0.000000e+00
## [3,] 0.000000e+00 0 80 0.000000e+00
## [4,] 4.235165e-14 0 0 2.000000e+01
##
## $code
## [1] 1
##
## $iterations
## [1] 1
## BUT ... it looks like nlminb is NOT using a true Newton-type method
t1nlminb <- nlminb(x0, fnt, gradient=grt, hessian=hesst, fscale=3.0,
control=list(trace=0))
print(t1nlminb)
## $par
## [1] -4.043175e-174 0.000000e+00 -3.234540e-173 -1.293816e-172
##
## $objective
## [1] 0
##
## $convergence
## [1] 0
##
## $iterations
## [1] 13
##
## $evaluations
## function gradient
## 15 13
##
## $message
## [1] "relative convergence (4)"
# and call them from optimx (i.e., test this gives same results)
t1so <- optimr(x0, fnt, grt, hess=hesst, method="snewton", fscale=3.0,
control=list(trace=0))
proptimr(t1so)
## Result t1so proposes optimum function value = 0 at parameters
## [1] 0 0 0 0
## After 2 fn evals, and 2 gr evals
## Termination code is 0 : Small gradient norm
## ---------------------------------------
t1nlmo <- optimr(x0, fnt, grt, hess=hesst, method="nlm", fscale=3.0,
control=list(trace=0))
proptimr(t1nlmo)
## Result t1nlmo proposes optimum function value = 1.232595e-29 at parameters
## [1] 0.000000e+00 2.220446e-16 4.440892e-16 0.000000e+00
## After NA fn evals, and 1 gr evals
## Termination code is 0 : nlm: Convergence indicator (code) = 1
## ---------------------------------------
tst <- try(t1nlminbo <- optimr(x0, fnt, grt, hess=hesst, method="nlminb",
fscale=3.0, control=list(trace=0)))
if (class(tst) == "try-error"){
cat("try-error on attempt to run nlminb in optimr()\n")
} else { proptimr(t1nlminbo) }
## Result t1nlminbo proposes optimum function value = 0 at parameters
## [1] -4.043175e-174 0.000000e+00 -3.234540e-173 -1.293816e-172
## After 15 fn evals, and 13 gr evals
## Termination code is 0 : relative convergence (4)
## ---------------------------------------
From the number of function and gradient evaluations, it appears nlminb() is not
using the Hessian information. Note that the snewton() and snewtonm() functions
return count information for iterations and hessian evaluations, but these are not
part of the standard optim() (and thus optimr()) result objects.
require(optimx)
#Rosenbrock banana valley function
f <- function(x){
return(100*(x[2] - x[1]*x[1])^2 + (1-x[1])^2)
}
#gradient
gr <- function(x){
return(c(-400*x[1]*(x[2] - x[1]*x[1]) - 2*(1-x[1]), 200*(x[2] - x[1]*x[1])))
}
#Hessian
h <- function(x) {
a11 <- 2 - 400*x[2] + 1200*x[1]*x[1]; a21 <- -400*x[1]
return(matrix(c(a11, a21, a21, 200), 2, 2))
}
x0 <- c(-1.2, 1)
# sink("mbrn1-170408.txt", split=TRUE)
t1 <- snewton(x0, fn=f, gr=gr, hess=h, control=list(trace=0))
print(t1)
## $par
## [1] 1 1
##
## $value
## [1] 0
##
## $grad
## [1] 0 0
##
## $Hess
## [,1] [,2]
## [1,] 802 -400
## [2,] -400 200
##
## $counts
## $counts$niter
## [1] 25
##
## $counts$nfn
## [1] 33
##
## $counts$ngr
## [1] 25
##
## $counts$nhess
## [1] 24
##
##
## $convcode
## [1] 0
##
## $message
## [1] "Small gradient norm"
# we can also use nlm and nlminb
fght <- function(x){
## combine f, g and h into single function for nlm
ff <- f(x)
gg <- gr(x)
hh <- h(x)
attr(ff, "gradient") <- gg
attr(ff, "hessian") <- hh
ff
}
# COULD TRY: t1nlm <- nlm(fght, x0, hessian=TRUE, print.level=2, iterlim=10000)
t1nlmo <- optimr(x0, f, gr, hess=h, method="nlm", control=list(trace=0))
proptimr(t1nlmo)
## Result t1nlmo proposes optimum function value = 1.127026e-17 at parameters
## [1] 1 1
## After NA fn evals, and 24 gr evals
## Termination code is 0 : nlm: Convergence indicator (code) = 1
## ---------------------------------------
t1so <- optimr(x0, f, gr, hess=h, method="snewton", control=list(trace=0))
proptimr(t1so)
## Result t1so proposes optimum function value = 0 at parameters
## [1] 1 1
## After 33 fn evals, and 25 gr evals
## Termination code is 0 : Small gradient norm
## ---------------------------------------
t1smo <- optimr(x0, f, gr, hess=h, method="snewtonm", control=list(trace=0))
## Small gradient
proptimr(t1smo)
## Result t1smo proposes optimum function value = 0 at parameters
## [1] 1 1
## After 46 fn evals, and 28 gr evals
## Termination code is 0 : snewtonm: Normal exit
## ---------------------------------------
## Again, nlminb probably not using hessian
tst <- try(t1nlminbo <- optimr(x0, f, gr, hess=h, method="nlminb",
control=list(trace=0)))
if (class(tst) == "try-error"){
cat("try-error on attempt to run nlminb in optimr()\n")
} else { proptimr(t1nlminbo) }
## Result t1nlminbo proposes optimum function value = 0 at parameters
## [1] 1 1
## After 33 fn evals, and 26 gr evals
## Termination code is 0 : X-convergence (3)
## ---------------------------------------
For nlm() the “standard” start takes more than 100 iterations and returns
a non-optimal solution.
#Example: Wood function
#
wood.f <- function(x){
res <- 100*(x[1]^2-x[2])^2+(1-x[1])^2+90*(x[3]^2-x[4])^2+(1-x[3])^2+
10.1*((1-x[2])^2+(1-x[4])^2)+19.8*(1-x[2])*(1-x[4])
return(res)
}
#gradient:
wood.g <- function(x){
g1 <- 400*x[1]^3-400*x[1]*x[2]+2*x[1]-2
g2 <- -200*x[1]^2+220.2*x[2]+19.8*x[4]-40
g3 <- 360*x[3]^3-360*x[3]*x[4]+2*x[3]-2
g4 <- -180*x[3]^2+200.2*x[4]+19.8*x[2]-40
return(c(g1,g2,g3,g4))
}
#hessian:
wood.h <- function(x){
h11 <- 1200*x[1]^2-400*x[2]+2; h12 <- -400*x[1]; h13 <- h14 <- 0
h22 <- 220.2; h23 <- 0; h24 <- 19.8
h33 <- 1080*x[3]^2-360*x[4]+2; h34 <- -360*x[3]
h44 <- 200.2
H <- matrix(c(h11,h12,h13,h14,h12,h22,h23,h24,
h13,h23,h33,h34,h14,h24,h34,h44),ncol=4)
return(H)
}
wood.fgh <- function(x){
fval <- wood.f(x)
gval <- wood.g(x)
hval <- wood.h(x)
attr(fval,"gradient") <- gval
attr(fval,"hessian")<- hval
fval
}
#################################################
x0 <- c(-3,-1,-3,-1) # Wood standard start
require(optimx)
# In 100 iterations, not converged
t1nlm <- nlm(wood.fgh, x0, print.level=0)
print(t1nlm)
## $minimum
## [1] 7.874467
##
## $estimate
## [1] -1.0316067 1.0740774 -0.9012710 0.8239815
##
## $gradient
## [1] 0.007517820 -0.015797320 -0.008937001 0.015745111
##
## $code
## [1] 4
##
## $iterations
## [1] 100
# But both newton approaches do work
wd <- snewton(x0, fn=wood.f, gr=wood.g, hess=wood.h, control=list(trace=0))
print(wd)
## $par
## [1] 1 1 1 1
##
## $value
## [1] 1.679904e-29
##
## $grad
## [1] 5.684342e-14 -1.421085e-14 5.817569e-14 1.421085e-14
##
## $Hess
## [,1] [,2] [,3] [,4]
## [1,] 802 -400.0 0 0.0
## [2,] -400 220.2 0 19.8
## [3,] 0 0.0 722 -360.0
## [4,] 0 19.8 -360 200.2
##
## $counts
## $counts$niter
## [1] 69
##
## $counts$nfn
## [1] 120
##
## $counts$ngr
## [1] 69
##
## $counts$nhess
## [1] 69
##
##
## $convcode
## [1] 93
##
## $message
## [1] "No progress in linesearch!"
wdm <- snewtonm(x0, fn=wood.f, gr=wood.g, hess=wood.h, control=list(trace=0))
print(wdm)
## $par
## [1] 1 1 1 1
##
## $value
## [1] 1.082712e-29
##
## $grad
## [1] -5.462297e-14 -7.105427e-15 -5.817569e-14 0.000000e+00
##
## $Hess
## [,1] [,2] [,3] [,4]
## [1,] 802 -400.0 0 0.0
## [2,] -400 220.2 0 19.8
## [3,] 0 0.0 722 -360.0
## [4,] 0 19.8 -360 200.2
##
## $counts
## $counts$niter
## [1] 95
##
## $counts$nfn
## [1] 94
##
## $counts$ngr
## [1] 53
##
## $counts$nhess
## [1] 53
##
##
## $convcode
## [1] 0
##
## $message
## [1] "snewtonm: Normal exit"
## AND again nlminb not likely using hessian information
## t1nlminb <- nlminb(x0, wood.f, gradient=wood.g, hess=wood.h, control=list(trace=0))
## print(t1nlminb)
# and call them from optimx (i.e., test this gives same results)
# But optimr uses a larger iteration limit, and gets to solution
t1nlmo <- optimr(x0, wood.f, wood.g, hess=wood.h, method="nlm", control=list(trace=0))
proptimr(t1nlmo)
## Result t1nlmo proposes optimum function value = 1.00494e-16 at parameters
## [1] 1 1 1 1
## After NA fn evals, and 335 gr evals
## Termination code is 0 : nlm: Convergence indicator (code) = 1
## ---------------------------------------
tst<-try(t1nlminbo <- optimr(x0, wood.f, wood.g, hess=wood.h, method="nlminb", control=list(trace=0)))
if (class(tst) == "try-error"){
cat("try-error on attempt to run nlminb in optimr()\n")
} else { proptimr(t1nlminbo) }
## Result t1nlminbo proposes optimum function value = 1.841497e-30 at parameters
## [1] 1 1 1 1
## After 54 fn evals, and 44 gr evals
## Termination code is 0 : X-convergence (3)
## ---------------------------------------
There are several generalizations of the Rosenbrock function (??ref)
# genrosa function code -- attempts to match the rosenbrock at gs=100 and x=c(-1.2,1)
genrosa.f<- function(x, gs=NULL){ # objective function
## One generalization of the Rosenbrock banana valley function (n parameters)
n <- length(x)
if(is.null(gs)) { gs=100.0 }
# Note do not at 1.0 so min at 0
fval<-sum (gs*(x[1:(n-1)]^2 - x[2:n])^2 + (x[1:(n-1)] - 1)^2)
}
genrosa.g <- function(x, gs=NULL){
# vectorized gradient for genrose.f
# Ravi Varadhan 2009-04-03
n <- length(x)
if(is.null(gs)) { gs=100.0 }
gg <- as.vector(rep(0, n))
tn <- 2:n
tn1 <- tn - 1
z1 <- x[tn] - x[tn1]^2
z2 <- 1 - x[tn1]
# f = gs*z1*z1 + z2*z2
gg[tn] <- 2 * (gs * z1)
gg[tn1] <- gg[tn1] - 4 * gs * x[tn1] * z1 - 2 *z2
return(gg)
}
genrosa.h <- function(x, gs=NULL) { ## compute Hessian
if(is.null(gs)) { gs=100.0 }
n <- length(x)
hh<-matrix(rep(0, n*n),n,n)
for (i in 2:n) {
z1<-x[i]-x[i-1]*x[i-1]
# z2<-1.0 - x[i-1]
hh[i,i]<-hh[i,i]+2.0*(gs+1.0)
hh[i-1,i-1]<-hh[i-1,i-1]-4.0*gs*z1-4.0*gs*x[i-1]*(-2.0*x[i-1])
hh[i,i-1]<-hh[i,i-1]-4.0*gs*x[i-1]
hh[i-1,i]<-hh[i-1,i]-4.0*gs*x[i-1]
}
return(hh)
}
require(optimx)
cat("Generalized Rosenbrock tests\n")
## Generalized Rosenbrock tests
cat("original n and x0")
## original n and x0
x0 <- c(-1.2, 1)
solorig <- snewton(x0, genrosa.f, genrosa.g, genrosa.h)
print(solorig)
## $par
## [1] 1 1
##
## $value
## [1] 2.972526e-28
##
## $grad
## [1] 5.462297e-14 -4.440892e-14
##
## $Hess
## [,1] [,2]
## [1,] 800 -400
## [2,] -400 202
##
## $counts
## $counts$niter
## [1] 128
##
## $counts$nfn
## [1] 143
##
## $counts$ngr
## [1] 128
##
## $counts$nhess
## [1] 128
##
##
## $convcode
## [1] 92
##
## $message
## [1] "No progress before linesearch!"
print(eigen(solorig$Hess)$values)
## [1] 1000.400641 1.599359
solorigm <- snewtonm(x0, genrosa.f, genrosa.g, genrosa.h)
print(solorigm)
## $par
## [1] 1 1
##
## $value
## [1] 1.232595e-30
##
## $grad
## [1] -2.220446e-15 0.000000e+00
##
## $Hess
## [,1] [,2]
## [1,] 800 -400
## [2,] -400 202
##
## $counts
## $counts$niter
## [1] 152
##
## $counts$nfn
## [1] 151
##
## $counts$ngr
## [1] 132
##
## $counts$nhess
## [1] 132
##
##
## $convcode
## [1] 0
##
## $message
## [1] "snewtonm: Normal exit"
print(eigen(solorigm$Hess)$values)
## [1] 1000.400641 1.599359
cat("Start with 50 values of pi and scale factor 10\n")
## Start with 50 values of pi and scale factor 10
x0 <- rep(pi, 50)
sol50pi <- optimr(x0, genrosa.f, genrosa.g, genrosa.h, method="snewton", gs=10)
proptimr(sol50pi)
## Result sol50pi proposes optimum function value = 6.108742e-29 at parameters
## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [39] 1 1 1 1 1 1 1 1 1 1 1 1
## After 145 fn evals, and 145 gr evals
## Termination code is 92 : No progress before linesearch!
## ---------------------------------------
hhi <- genrosa.h(sol50pi$par, gs=10)
print(eigen(hhi)$values)
## [1] 181.84200 181.36863 180.58176 179.48449 178.08116 176.37730 174.37964
## [8] 172.09607 169.53560 166.70834 163.62545 160.29911 156.74243 152.96948
## [15] 148.99513 144.83509 140.50578 136.02429 131.40832 126.67610 121.84632
## [22] 116.93804 111.97066 106.96381 101.93725 96.91085 91.90447 86.93791
## [29] 82.03080 77.20253 72.47223 67.85859 63.37989 59.05387 54.89766
## [36] 50.92776 47.15992 43.60907 40.28933 37.21385 34.39481 31.84332
## [43] 29.56937 27.58175 25.88797 24.49427 23.40556 22.62547 22.15648
## [50] 2.00000
sol50pim <- optimr(x0, genrosa.f, genrosa.g, genrosa.h, method="snewtonm", gs=10)
proptimr(sol50pim)
## Result sol50pim proposes optimum function value = 1.528418e-30 at parameters
## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [39] 1 1 1 1 1 1 1 1 1 1 1 1
## After 157 fn evals, and 152 gr evals
## Termination code is 0 : snewtonm: Normal exit
## ---------------------------------------
hhm <- genrosa.h(sol50pim$par, gs=10)
print(eigen(hhm)$values)
## [1] 181.84200 181.36863 180.58176 179.48449 178.08116 176.37730 174.37964
## [8] 172.09607 169.53560 166.70834 163.62545 160.29911 156.74243 152.96948
## [15] 148.99513 144.83509 140.50578 136.02429 131.40832 126.67610 121.84632
## [22] 116.93804 111.97066 106.96381 101.93725 96.91085 91.90447 86.93791
## [29] 82.03080 77.20253 72.47223 67.85859 63.37989 59.05387 54.89766
## [36] 50.92776 47.15992 43.60907 40.28933 37.21385 34.39481 31.84332
## [43] 29.56937 27.58175 25.88797 24.49427 23.40556 22.62547 22.15648
## [50] 2.00000
This problem is described in @cnm79. It has various nasty properties. Note that one starting point causes
failure of the snewton() optimizer.
## Optimization test function HOBBS
## ?? refs (put in .doc??)
## Nash and Walker-Smith (1987, 1989) ...
require(optimx)
hobbs.f<- function(x){ # # Hobbs weeds problem -- function
if (abs(12*x[3]) > 500) { # check computability
fbad<-.Machine$double.xmax
return(fbad)
}
res<-hobbs.res(x)
f<-sum(res*res)
}
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)
}
hobbs.rsd<-function(x) { # Jacobian second derivative
rsd<-array(0.0, c(12,3,3))
t<-1:12
yy<-exp(-x[3]*t)
zz<-1.0/(1+x[2]*yy)
rsd[t,1,1]<- 0.0
rsd[t,2,1]<- -yy*zz*zz
rsd[t,1,2]<- -yy*zz*zz
rsd[t,2,2]<- 2.0*x[1]*yy*yy*zz*zz*zz
rsd[t,3,1]<- t*x[2]*yy*zz*zz
rsd[t,1,3]<- t*x[2]*yy*zz*zz
rsd[t,3,2]<- t*x[1]*yy*zz*zz*(1-2*x[2]*yy*zz)
rsd[t,2,3]<- t*x[1]*yy*zz*zz*(1-2*x[2]*yy*zz)
## rsd[t,3,3]<- 2*t*t*x[1]*x[2]*x[2]*yy*yy*zz*zz*zz
rsd[t,3,3]<- -t*t*x[1]*x[2]*yy*zz*zz*(1-2*yy*zz*x[2])
return(rsd)
}
hobbs.h <- function(x) { ## compute Hessian
# cat("Hessian not yet available\n")
# return(NULL)
H<-matrix(0,3,3)
res<-hobbs.res(x)
jj<-hobbs.jac(x)
rsd<-hobbs.rsd(x)
## H<-2.0*(t(res) %*% rsd + t(jj) %*% jj)
for (j in 1:3) {
for (k in 1:3) {
for (i in 1:12) {
H[j,k]<-H[j,k]+res[i]*rsd[i,j,k]
}
}
}
H<-2*(H + t(jj) %*% jj)
return(H)
}
require(optimx)
x0 <- c(200, 50, .3)
cat("Start for Hobbs:")
## Start for Hobbs:
print(x0)
## [1] 200.0 50.0 0.3
solx0 <- snewton(x0, hobbs.f, hobbs.g, hobbs.h)
## Note that we exceed count limit, but have answer
print(solx0)
## $par
## [1] 196.1862618 49.0916395 0.3135697
##
## $value
## [1] 2.587277
##
## $grad
## [1] -4.440892e-16 2.109424e-15 -3.979039e-13
##
## $Hess
## [,1] [,2] [,3]
## [1,] 1.265461 -3.256125 1602.105
## [2,] -3.256125 8.627095 -4095.206
## [3,] 1602.105263 -4095.206388 2043434.033
##
## $counts
## $counts$niter
## [1] 10
##
## $counts$nfn
## [1] 10
##
## $counts$ngr
## [1] 10
##
## $counts$nhess
## [1] 10
##
##
## $convcode
## [1] 92
##
## $message
## [1] "No progress before linesearch!"
print(eigen(solx0$Hess)$values)
## [1] 2.043443e+06 4.249248e-01 4.413953e-03
## Note that we exceed count limit, but have answer
## Setting relative check offset larger gets quicker convergence
solx0a <- snewton(x0, hobbs.f, hobbs.g, hobbs.h, control=list(offset=1000.))
print(solx0a)
## $par
## [1] 196.1862618 49.0916395 0.3135697
##
## $value
## [1] 2.587277
##
## $grad
## [1] -4.440892e-16 2.109424e-15 -3.979039e-13
##
## $Hess
## [,1] [,2] [,3]
## [1,] 1.265461 -3.256125 1602.105
## [2,] -3.256125 8.627095 -4095.206
## [3,] 1602.105263 -4095.206388 2043434.033
##
## $counts
## $counts$niter
## [1] 10
##
## $counts$nfn
## [1] 10
##
## $counts$ngr
## [1] 10
##
## $counts$nhess
## [1] 10
##
##
## $convcode
## [1] 92
##
## $message
## [1] "No progress before linesearch!"
x1s <- c(100, 10, .1)
cat("Start for Hobbs:")
## Start for Hobbs:
print(x1s)
## [1] 100.0 10.0 0.1
solx1s <- snewton(x1s, hobbs.f, hobbs.g, hobbs.h, control=list(trace=0))
print(solx1s)
## $par
## [1] 196.1862618 49.0916395 0.3135697
##
## $value
## [1] 2.587277
##
## $grad
## [1] 1.665335e-16 -2.220446e-16 2.273737e-13
##
## $Hess
## [,1] [,2] [,3]
## [1,] 1.265461 -3.256125 1602.105
## [2,] -3.256125 8.627095 -4095.206
## [3,] 1602.105263 -4095.206388 2043434.033
##
## $counts
## $counts$niter
## [1] 36
##
## $counts$nfn
## [1] 45
##
## $counts$ngr
## [1] 36
##
## $counts$nhess
## [1] 36
##
##
## $convcode
## [1] 92
##
## $message
## [1] "No progress before linesearch!"
print(eigen(solx1s$Hess)$values)
## [1] 2.043443e+06 4.249248e-01 4.413953e-03
solx1m <- snewton(x1s, hobbs.f, hobbs.g, hobbs.h, control=list(trace=0))
print(solx1m)
## $par
## [1] 196.1862618 49.0916395 0.3135697
##
## $value
## [1] 2.587277
##
## $grad
## [1] 1.665335e-16 -2.220446e-16 2.273737e-13
##
## $Hess
## [,1] [,2] [,3]
## [1,] 1.265461 -3.256125 1602.105
## [2,] -3.256125 8.627095 -4095.206
## [3,] 1602.105263 -4095.206388 2043434.033
##
## $counts
## $counts$niter
## [1] 36
##
## $counts$nfn
## [1] 45
##
## $counts$ngr
## [1] 36
##
## $counts$nhess
## [1] 36
##
##
## $convcode
## [1] 92
##
## $message
## [1] "No progress before linesearch!"
print(eigen(solx1m$Hess)$values)
## [1] 2.043443e+06 4.249248e-01 4.413953e-03
cat("Following test fails in snewton with ERROR -- Why?\n")
## Following test fails in snewton with ERROR -- Why?
x3 <- c(1, 1, 1)
cat("Start for Hobbs:")
## Start for Hobbs:
print(x3)
## [1] 1 1 1
ftest <- try(solx3 <- snewton(x3, hobbs.f, hobbs.g, hobbs.h, control=list(trace=0)))
if (class(ftest) != "try-error") {
print(solx3)
print(eigen(solx3$Hess)$values)
}
## $par
## [1] 1 1 1
##
## $value
## [1] 23520.58
##
## $grad
## [1] -824.042084 4.764888 -11.025384
##
## $Hess
## [,1] [,2] [,3]
## [1,] 22.321687 4.150703 -9.853740
## [2,] 4.150703 -1.275310 -9.255147
## [3,] -9.853740 -9.255147 33.506883
##
## $counts
## $counts$niter
## [1] 1
##
## $counts$nfn
## [1] 2
##
## $counts$ngr
## [1] 1
##
## $counts$nhess
## [1] 1
##
##
## $convcode
## [1] 9999
##
## $message
## [1] "snewton: New function value infinite"
##
## [1] 41.618914 16.635191 -3.700846
cat("But Marquardt variant succeeds\n")
## But Marquardt variant succeeds
solx3m <- snewtonm(x3, hobbs.f, hobbs.g, hobbs.h, control=list(trace=0))
print(solx3m)
## $par
## [1] 196.1862617 49.0916395 0.3135697
##
## $value
## [1] 2.587277
##
## $grad
## [1] -8.157010e-09 2.019953e-08 -1.026682e-05
##
## $Hess
## [,1] [,2] [,3]
## [1,] 1.265461 -3.256125 1602.105
## [2,] -3.256125 8.627095 -4095.206
## [3,] 1602.105263 -4095.206388 2043434.032
##
## $counts
## $counts$niter
## [1] 59
##
## $counts$nfn
## [1] 58
##
## $counts$ngr
## [1] 26
##
## $counts$nhess
## [1] 26
##
##
## $convcode
## [1] 0
##
## $message
## [1] "snewtonm: Normal exit"
print(eigen(solx3m$Hess)$values)
## [1] 2.043443e+06 4.249248e-01 4.413953e-03
# we can also use nlm and nlminb and call them from optimx