Boxed Region Transformation

Description

Transformation of a box/bound constrained region to an unconstrained one.

Usage

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transfinite(lower, upper, n = length(lower))

Arguments

lower, upper

lower and upper box/bound constraints.

n

length of upper, lower if both are scalars, to which they get repeated.

Details

Transforms a constraint region in n-dimensional space bijectively to the unconstrained R^n space, applying a atanh resp. exp transformation to each single variable that is bound constraint.

It provides two functions, h: B = []x...x[] --> R^n and its inverse hinv. These functions can, for example, be used to add box/bound constraints to a constrained optimization problem that is to be solved with a (nonlinear) solver not allowing constraints.

Value

Returns to functions as components h and hinv of a list.

Note

Based on an idea of Ravi Varadhan, intrinsically used in his implementation of Nelder-Mead in the ‘dfoptim’ package.

For positivity constraints, x>=0, this approach is considered to be numerically more stable than x-->exp(x) or x-->x^2.

Examples

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lower <- c(-Inf, 0,   0)
upper <- c( Inf, 0.5, 1)
Tf <- transfinite(lower, upper)
h <- Tf$h; hinv <- Tf$hinv

## Not run: 
##  Solve Rosenbrock with one variable restricted
rosen <- function(x) {
    n <- length(x)
    x1 <- x[2:n]; x2 <- x[1:(n-1)]
    sum(100*(x1-x2^2)^2 + (1-x2)^2)
}
f  <- function(x) rosen(hinv(x))    # f must be defined on all of R^n
x0 <- c(0.1, 0.1, 0.1)              # starting point not on the boundary!
nm <- nelder_mead(h(x0), f)         # unconstraint Nelder-Mead
hinv(nm$xmin); nm$fmin              # box/bound constraint solution
# [1] 0.7085596 0.5000000 0.2500004
# [1] 0.3353605

## End(Not run)