goldensection: Golden Section Method.
In brunasqz/NonlinearOpMethods: Nonlinear Optimization Methods

Description Usage Arguments Value About the parameter phi References See Also Examples

goldensection is a line search method that computes the optimum of a univariate function over an interval [a,b] where it is known to be unimodal.

1	goldensection(phi, a = 0, b = 1, eps.alpha = 1e-06)

`phi`	A univariate function.
`a, b`	Numbers, initial interval where the function is unimodal.
`eps.alpha`	A number, a tolerance to stop the optimization when the current interval becomes smaller `eps`.

Step length alpha where phi attains its minimum in [a,b] (assuming unimodality).

phi should be either an originally univariate function phi(alpha), or the result of a multi-variate one over a fixed interval phi(alpha) := f(x + alpha*searchD), which should come from the univariate_f function.

Rao, Singiresu S.; Engineering Optimization Theory and and Practice, 4th ed., page 267.
Ramirez, Jaime A.; Campelo, Felipe; Guimaraes, Frederico G.; Batista, Lucas S.; Takahashi, Ricardo H. C.; Notas de aula de Otimizacao, pages 10:13.

The documentation of the function univariate_f in this package.

# A univariate function
phi <- function(alpha)
{
  return (alpha^5 - 5*alpha^3 - 20*alpha + 5)
}
# Plot the function for convenience over the interval [-5,5]
plot(phi, -5, 5)
# Compute the optimum in the interval [0,5] where it is unimodal
alpha.opt1 <- goldensection(phi, a = 0, b = 5) #approx 2
# If it is changed to [-5,0] where it is not unimodal, an extreme is returned
alpha.opt2 <- goldensection(phi, a = -5, b = 0) #return -5

# Considering a multi-variate function, assume given the current point x and
# the search direction searchD
f <- function (x)
{
  i <- seq(1, length(x))
  return (sum( (x - i)^2 ))
}
x <- c(0,0,0,0) #current point
searchD <- c(1,2,3,4) #search direction
univariate_f <- function(f, x, searchD) {
phi <- f(x)
function(alpha) {
 phi <- f(x + alpha * searchD)

 return(phi)
 }
}
phi <- univariate_f(f, x, searchD) #return univariate version
# Compute optimum over the line x + alpha*searchD in [0,2]
alpha.opt <- goldensection(phi, a = 0, b = 2) #approx 1