R/30_broyden_tri.R

In jlmelville/funconstrain: Functions for Testing Unconstrained Numerical Optimization

#' Broyden Tridiagonal Function
#'
#' Test function 30 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} variable, number of summand
#'   functions \code{m = n}.
#'   \item Minima: \code{f = 0}.
#' }
#'
#' The number of parameters, \code{n}, in the objective function is not
#' specified when invoking this function. It is implicitly set by the length of
#' the parameter vector passed to the objective and gradient functions that this
#' function creates. See the 'Examples' section.
#'
#' @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} Function returning the standard starting point, given
#'   \code{n}, the number of variables desired.
#'   \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}
#'
#' Broyden, C. G. (1965).
#' A class of methods for solving nonlinear simultaneous equations.
#' \emph{Mathematics of computation}, \emph{19}(92), 577-593.
#' \doi{doi.org/10.2307/2003941}
#'
#' @examples
#' btri <- broyden_tri()
#' # 6 variable problem using the standard starting point
#' x0_6 <- btri$x0(6)
#' res_6 <- stats::optim(x0_6, btri$fn, btri$gr, method = "L-BFGS-B")
#' # Standing starting point with 8 variables
#' res_8 <- stats::optim(btri$x0(8), btri$fn, btri$gr, method = "L-BFGS-B")
#' # Create your own 4 variable starting point
#' res_4 <- stats::optim(c(0.1, 0.2, 0.3, 0.4), btri$fn, btri$gr,
#'                       method = "L-BFGS-B")
#' @export
broyden_tri <- function() {
  list(
    fn = function(par) {
      n <- length(par)
      if (n < 1) {
        stop("Broyden Tridiagonal: n must be positive")
      }

      fi <- (3 - 2 * par) * par + 1
      fi[1:(n - 1)] <- fi[1:(n - 1)] - 2 * par[2:n]
      fi[2:n] <- fi[2:n] - par[1:(n - 1)]
      sum(fi * fi)
    },
    gr = function(par) {
      n <- length(par)
      if (n < 1) {
        stop("Broyden Tridiagonal: n must be positive")
      }

      fi <- (3 - 2 * par) * par + 1
      fi[1:(n - 1)] <- fi[1:(n - 1)] - 2 * par[2:n]
      fi[2:n] <- fi[2:n] - par[1:(n - 1)]

      grad <- 2 * fi * (3 - 4 * par)
      grad[1:(n - 1)] <- grad[1:(n - 1)] - 2 * fi[2:n]
      grad[2:n] <- grad[2:n] - 4 * fi[1:(n - 1)]

      grad
    },
    he = function(x) { 
       n <- length(x)
       h <- matrix(0.0, nrow=n, ncol=n)
#       ! For i <- 1
       t  <- ( 3.0 - 2.0*x[1] )*x[1] - 2.0*x[2] + 1.0
       t1 <- 3.0 - 4.0*x[1]
       h[1,1] <-   2.0*( t1 ^ 2 - 4.0*t )
       h[1,2] <- - 4.0*t1
       h[2,2] <-   8.0

       for (i in 2:(n-1)) {
          t  <- ( 3.0 - 2.0*x[i] )*x[i] - x[i-1] - 2.0*x[i+1] + 1.0
          t1 <- 3.0 - 4.0*x[i]
          h[i-1,i-1] <- h[i-1,i-1] + 2.0
          h[i-1,i  ] <- h[i-1,i  ] - 2.0*t1
          h[i,  i  ] <- h[i,  i  ] + 2.0*( t1 ^ 2 - 4.0*t )
          h[i-1,i+1] <- h[i-1,i+1] + 4.0
          h[i  ,i+1] <- h[i  ,i+1] - 4.0*t1
          h[i+1,i+1] <- h[i+1,i+1] + 8.0
       }

#       ! For i <- n
       t  <- ( 3.0 - 2.0*x[n] )*x[n] - x[n-1] + 1.0
       t1 <- 3.0 - 4.0*x[n]
       h[n-1,n-1] <- h[n-1, n-1] + 2.0
       h[n-1,  n] <- h[n-1, n  ] - 2.0*t1
       h[  n,  n] <- h[ n,  n  ] + 2.0*( t1 ^ 2 - 4.0*t )

       for (j in 1:(n-1)) { # symmetrize
         for (k in (j+1):n) {
           h[k,j] <- h[j,k]        
         }
       }
       h
    },

    fg = function(par) {
      n <- length(par)
      if (n < 1) {
        stop("Broyden Tridiagonal: n must be positive")
      }

      fi <- (3 - 2 * par) * par + 1
      fi[1:(n - 1)] <- fi[1:(n - 1)] - 2 * par[2:n]
      fi[2:n] <- fi[2:n] - par[1:(n - 1)]
      fsum <- sum(fi * fi)

      grad <- 2 * fi * (3 - 4 * par)
      grad[1:(n - 1)] <- grad[1:(n - 1)] - 2 * fi[2:n]
      grad[2:n] <- grad[2:n] - 4 * fi[1:(n - 1)]

      list(
        fn = fsum,
        gr = grad
      )
    },
    x0 = function(n = 40) {
      if (n < 1) {
        stop("Broyden Tridiagonal: n must be positive")
      }
      rep(-1, n)
    },
    fmin = 0,
    xmin = c(-0.5648284, -0.6662737, -0.6609170, -0.5950500, -0.4162011) # n = 5 case
  )
}