optimizeR: High Precisione One-Dimensional Optimization

Description

optimizeR searches the intervalfrom lower to upper for a minimum of the function f with respect to its first argument.

Usage

optimizeR(f, lower, upper, ..., tol = 1e-20,
          method = c("Brent", "GoldenRatio"),
          maximum = FALSE,
          precFactor = 2.0, precBits = -log2(tol) * precFactor,
          maxiter = 1000, trace = FALSE)

Arguments

the function to be optimized. f(x) must work in Rmpfr arithmetic for optimizer() to make sense. The function is either minimized or maximized over its first argument depending on the value

...

additional named or unnamed arguments to be passed to f.

lower

the lower end point of the interval to be searched.

upper

the upper end point of the interval to be searched.

tol

the desired accuracy, typically higher than double precision, i.e., tol < 2e-16.

method

character string specifying the optimization method.

maximum

logical indicating if $f()$ should be maximized or minimized (the default).

precFactor

only for default precBits construction: a factor to multiply with the number of bits directly needed for tol.

precBits

number of bits to be used for mpfr numbers used internally.

maxiter

maximal number of iterations to be used.

trace

integer or logical indicating if and how iterations should be monitored; if an integer $k$, print every $k$-th iteration.

Value

A list with components minimum (or maximum) and objective which give the location of the minimum (or maximum) and the value of the function at that point; iter specifiying the number of iterations, the logical convergence indicating if the iterations converged and estim.prec which is an estimate or an upper bound of the final precision (in $x$). method the string of the method used.

Details

[object Object],[object Object]

Examples

Run this code

iG5 <- function(x) -exp(-(x-5)^2/2)
curve(iG5, 0, 10, 200)
o.dp  <- optimize (iG5, c(0, 10)) #->  5 of course
oM.gs <- optimizeR(iG5, 0, 10, method="Golden")
oM.Br <- optimizeR(iG5, 0, 10, method="Brent", trace=TRUE)
oM.gs$min ; oM.gs$iter
oM.Br$min ; oM.Br$iter
(doExtras <- Rmpfr:::doExtras())
if(doExtras) {## more accuracy {takes a few seconds}
 oM.gs <- optimizeR(iG5, 0, 10, method="Golden", tol = 1e-70)
 oM.Br <- optimizeR(iG5, 0, 10,                  tol = 1e-70)
}
rbind(Golden = c(err = as.numeric(oM.gs$min -5), iter = oM.gs$iter),
      Brent  = c(err = as.numeric(oM.Br$min -5), iter = oM.Br$iter))

## ==> Brent is orders of magnitude more efficient !

## Testing on the sine curve with 40 correct digits:
sol <- optimizeR(sin, 2, 6, tol = 1e-40)
str(sol)
sol <- optimizeR(sin, 2, 6, tol = 1e-50,
                 precFactor = 3.0, trace = TRUE)
pi.. <- 2*sol$min/3
print(pi.., digits=51)
stopifnot(all.equal(pi.., Const("pi", 256), tol = 10*1e-50))

if(doExtras) { # considerably more expensive

## a harder one:
f.sq <- function(x) sin(x-2)^4 + sqrt(pmax(0,(x-1)*(x-4)))*(x-2)^2
curve(f.sq, 0, 4.5, n=1000)
msq <- optimizeR(f.sq, 0, 5, tol = 1e-50, trace=5)
str(msq) # ok
stopifnot(abs(msq$minimum - 2) < 1e-49)

## find the other local minimum: -- non-smooth ==> Golden-section is used
msq2 <- optimizeR(f.sq, 3.5, 5, tol = 1e-50, trace=10)
stopifnot(abs(msq2$minimum - 4) < 1e-49)

## and a local maximum:
msq3 <- optimizeR(f.sq, 3, 4, maximum=TRUE, trace=2)
stopifnot(abs(msq3$maximum - 3.57) < 1e-2)

}#end {doExtras}


##----- "impossible" one to get precisely ------------------------

ff <- function(x) exp(-1/(x-8)^2)
curve(exp(-1/(x-8)^2), -3, 13, n=1001)
(opt. <- optimizeR(function(x) exp(-1/(x-8)^2), -3, 13, trace = 5))
## -> close to 8 {but not very close!}
ff(opt.$minimum) # gives 0
if(doExtras) {
 ## try harder ... in vain ..
 str(opt1 <- optimizeR(ff, -3,13, tol = 1e-60, precFactor = 4))
 print(opt1$minimum, digits=20)
 ## still just  7.99998038 or 8.000036655 {depending on method}
}