R/tnewton.R

In nloptr: R Interface to NLopt

# Copyright (C) 2014 Hans W. Borchers. All Rights Reserved.
# This code is published under the L-GPL.
#
# File:   tnewton.R
# Author: Hans W. Borchers
# Date:   27 January 2014
#
# Wrapper to solve optimization problem using Preconditioned Truncated Newton.



#' Preconditioned Truncated Newton
#'
#' Truncated Newton methods, also calledNewton-iterative methods, solve an
#' approximating Newton system using a conjugate-gradient approach and are
#' related to limited-memory BFGS.
#'
#' Truncated Newton methods are based on approximating the objective with a
#' quadratic function and applying an iterative scheme such as the linear
#' conjugate-gradient algorithm.
#'
#' @param x0 starting point for searching the optimum.
#' @param fn objective function that is to be minimized.
#' @param gr gradient of function \code{fn}; will be calculated numerically if
#' not specified.
#' @param lower,upper lower and upper bound constraints.
#' @param precond logical; preset L-BFGS with steepest descent.
#' @param restart logical; restarting L-BFGS with steepest descent.
#' @param nl.info logical; shall the original NLopt info been shown.
#' @param control list of options, see \code{nl.opts} for help.
#' @param ... additional arguments passed to the function.
#'
#' @return List with components:
#'   \item{par}{the optimal solution found so far.}
#'   \item{value}{the function value corresponding to \code{par}.}
#'   \item{iter}{number of (outer) iterations, see \code{maxeval}.}
#'   \item{convergence}{integer code indicating successful completion (> 1)
#'     or a possible error number (< 0).}
#'   \item{message}{character string produced by NLopt and giving additional
#'     information.}
#'
#' @export tnewton
#'
#' @author Hans W. Borchers
#'
#' @note Less reliable than Newton's method, but can handle very large
#' problems.
#'
#' @seealso \code{\link{lbfgs}}
#'
#' @references R. S. Dembo and T. Steihaug, ``Truncated Newton algorithms for
#' large-scale optimization,'' Math. Programming 26, p. 190-212 (1982).
#'
#' @examples
#'
#' flb <- function(x) {
#'     p <- length(x)
#'     sum(c(1, rep(4, p-1)) * (x - c(1, x[-p])^2)^2)
#' }
#' # 25-dimensional box constrained: par[24] is *not* at boundary
#' S <- tnewton(rep(3, 25), flb, lower=rep(2, 25), upper=rep(4, 25),
#'                 nl.info = TRUE, control = list(xtol_rel=1e-8))
#' ## Optimal value of objective function:  368.105912874334
#' ## Optimal value of controls: 2  ...  2  2.109093  4
#'
tnewton <-
function(x0, fn, gr = NULL, lower = NULL, upper = NULL,
            precond = TRUE, restart = TRUE,
            nl.info = FALSE, control = list(), ...)
{
    opts <- nl.opts(control)
    if (precond) {
        if (restart)
            opts["algorithm"] <- "NLOPT_LD_TNEWTON_PRECOND_RESTART"
        else
            opts["algorithm"] <- "NLOPT_LD_TNEWTON_PRECOND"
    } else {
        if (restart)
            opts["algorithm"] <- "NLOPT_LD_TNEWTON_RESTART"
        else
            opts["algorithm"] <- "NLOPT_LD_TNEWTON"
    }

    fun <- match.fun(fn)
    fn  <- function(x) fun(x, ...)

    if (is.null(gr)) {
        gr <- function(x) nl.grad(x, fn)
    } else {
        .gr <- match.fun(gr)
        gr <- function(x) .gr(x, ...)
    }

    S0 <- nloptr(x0,
                eval_f = fn,
                eval_grad_f = gr,
                lb = lower,
                ub = upper,
                opts = opts)

    if (nl.info) print(S0)
    S1 <- list(par = S0$solution, value = S0$objective, iter = S0$iterations,
                convergence = S0$status, message = S0$message)
    return(S1)
}