R/02_freud_roth.R

In jlmelville/funconstrain: Functions for Testing Unconstrained Numerical Optimization

#' Freudenstein and Roth Function
#'
#' Test function 2 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 = 2}.
#'   \item Minima: \code{f = 0} at \code{(5, 4)},
#'   \code{f = 48.9842...} at \code{(11.41..., -0.8968...)}
#' }
#'
#' @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.
#' }
#' @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}
#'
#' Freudenstein, F., & Roth, B. (1963).
#' Numerical solution of systems of nonlinear equations.
#' \emph{Journal of the ACM (JACM)}, \emph{10}(4), 550-556.
#' \doi{doi.org/10.1145/321186.321200}
#'
#' @examples
#' fun <- freud_roth()
#' # 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
freud_roth <- function() {
  list(
    m = NA,
    fn = function(par) {
      x <- par[1]
      y <- par[2]

      f1 <- -13 + x + ((5 - y) * y - 2) * y
      f2 <- -29 + x + ((y + 1) * y - 14) * y

      f1 * f1 + f2 * f2
    },
    gr = function(par) {
      x <- par[1]
      y <- par[2]
      yy <- y * y
      yyyy <- yy * yy
      c(
        4 * x + 12 * y * y - 32 * y - 84,
        12 * yyyy * y - 40 * yyyy + 8 * yy * y - 240 * yy +
          (24 * x + 24) * y - 32 * x + 864
      )
    },
    he = function(par) {
       x1 <- par[1]
       x2 <- par[2]
       h <- matrix(NA, nrow=2, ncol=2)
       t1 <- - 13.0 + x1 + ( ( 5.0 - x2 )*x2 - 2.0 )*x2
       t2 <- - 29.0 + x1 + ( ( x2 + 1.0 )*x2 - 14.0 )*x2
       h[1,1] <- 4.0
       h[1,2] <- 8.0*( 3.0*x2 - 4.0 )
       h[2,2] <- 2.0*(                                                                     
            t1*( - 6.0*x2 + 10.0 ) + ( ( 10.0 - 3.0*x2 )*x2 - 2.0 ) ^ 2 + 
            t2*(   6.0*x2 +  2.0 ) + ( ( 3.0*x2 + 2.0 )*x2 - 14.0 ) ^ 2 )
       h[2,1] <- h[1,2]
       h
    },
    fg = function(par) {
      x <- par[1]
      y <- par[2]
      f1 <- -13 + x + ((5 - y) * y - 2) * y
      f2 <- -29 + x + ((y + 1) * y - 14) * y

      yy <- y * y
      yyyy <- yy * yy

      list(
        fn = f1 * f1 + f2 * f2,
        gr = c(2 * f1 + 2 * f2,
               12 * yyyy * y - 40 * yyyy + 8 * yy * y - 240 * yy +
                 (24 * x + 24) * y - 32 * x + 864)
      )
    },
    x0 = c(0.5, -2),
    fmin = 0,
    xmin = c(5, 4)
  )
}