tnewton {nloptr}

Preconditioned Truncated Newton

Description

Truncated Newton methods, also called Newton-iterative methods, solve an approximating Newton system using a conjugate-gradient approach and are related to limited-memory BFGS.

Usage

tnewton(
  x0,
  fn,
  gr = NULL,
  lower = NULL,
  upper = NULL,
  precond = TRUE,
  restart = TRUE,
  nl.info = FALSE,
  control = list(),
  ...
)

Arguments

x0

starting point for searching the optimum.

fn

objective function that is to be minimized.

gr

gradient of function fn; will be calculated numerically if not specified.

lower, upper

lower and upper bound constraints.

precond

logical; preset L-BFGS with steepest descent.

restart

logical; restarting L-BFGS with steepest descent.

nl.info

logical; shall the original NLopt info been shown.

control

list of options, see nl.opts for help.

...

additional arguments passed to the function.

Details

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.

Value

List with components:

par

the optimal solution found so far.

value

the function value corresponding to par.

iter

number of (outer) iterations, see maxeval.

convergence

integer code indicating successful completion (> 1) or a possible error number (< 0).

message

character string produced by NLopt and giving additional information.

Note

Less reliable than Newton's method, but can handle very large problems.

Author(s)

Hans W. Borchers

References

R. S. Dembo and T. Steihaug, “Truncated Newton algorithms for large-scale optimization,” Math. Programming 26, p. 190-212 (1982).

See Also

lbfgs

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, 25L), flb, lower = rep(2, 25L), upper = rep(4, 25L),
       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