R/04_brown_bs.R

In jlmelville/funconstrain: Functions for Testing Unconstrained Numerical Optimization

#' Brown Badly Scaled Function
#'
#' Test function 4 from the More', Garbow and Hillstrom paper.
#'
#' The objective function is the sum of \code{m} functions, each of \code{n}
#' parameters.
#'
#' \itemize{
#'   \item Dimensions: Number of parameters \code{n = 2}, number of summand
#'   functions \code{m = 3}.
#'   \item Minima: \code{f = 0} at \code{(1e6, 2e-6) },
#' }
#'
#' @return A list containing:
#' \itemize{
#'   \item \code{fn} Objective function which calculates the value given input
#'   parameter vector.
#'   \item \code{gr} Gradient function which calculates the gradient vector
#'   given input parameter vector.
#'   \item \code{he} If available, the hessian matrix (second derivatives)
#'   of the function w.r.t. the parameters at the given values.
#'   \item \code{fg} A function which, given the parameter vector, calculates
#'   both the objective value and gradient, returning a list with members
#'   \code{fn} and \code{gr}, respectively.
#'   \item \code{x0} Standard starting point.
#'   \item \code{fmin} reported minimum
#'   \item \code{xmin} parameters at reported minimum
#' }
#' @references
#' More', J. J., Garbow, B. S., & Hillstrom, K. E. (1981).
#' Testing unconstrained optimization software.
#' \emph{ACM Transactions on Mathematical Software (TOMS)}, \emph{7}(1), 17-41.
#' \doi{doi.org/10.1145/355934.355936}
#'
#' @examples
#' fun <- brown_bs()
#' # Optimize using the standard starting point
#' x0 <- fun$x0
#' res_x0 <- stats::optim(par = x0, fn = fun$fn, gr = fun$gr, method =
#' "L-BFGS-B")
#' # Use your own starting point
#' res <- stats::optim(c(0.1, 0.2), fun$fn, fun$gr, method = "L-BFGS-B")
#' @export
brown_bs <- function() {
  list(
    m = NA,
    fn = function(par) {
      x <- par[1]
      y <- par[2]

      f1 <- x - 1e6
      f2 <- y - 2e-6
      f3 <- x * y - 2

      f1 * f1 + f2 * f2 + f3 * f3
    },
    gr = function(par) {
      x <- par[1]
      y <- par[2]

      f3 <- x * y - 2

      c(
        2 * y * f3 + 2 * (x - 1e6),
        2 * x * f3 + 2 * (y - 2e-6)
      )
    },
    he = function(par) {
      x1 <- par[1]
      x2 <- par[2]
       h <- matrix(NA, nrow=2, ncol=2)
       h[1,1] <- 2.0*( 1.0 + x2 ^ 2 )
       h[1,2] <- 4.0*( x1*x2 - 1.0 )
       h[2,2] <- 2.0*( 1.0 + x1 ^ 2 )
       h[2,1] <- h[1,2]
       h
    },
    fg = function(par) {
      x <- par[1]
      y <- par[2]

      f1 <- x - 1e6
      f2 <- y - 2e-6
      f3 <- x * y - 2

      list(
        fn = f1 * f1 + f2 * f2 + f3 * f3,
        gr = c(
          2 * y * f3 + 2 * f1,
          2 * x * f3 + 2 * f2
        )
      )
    },
    x0 = c(1, 1),
    fmin = 0,
    xmin = c(1e6, 2e-6) 
  )
}