Preconditioned Truncated Newton

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.

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.

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.

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.

Note

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

References

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

Author

Hans W. Borchers

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))
#> 
#> Call:
#> nloptr(x0 = x0, eval_f = fn, eval_grad_f = gr, lb = lower, ub = upper, 
#>     opts = opts)
#> 
#> 
#> Minimization using NLopt version 2.7.1 
#> 
#> NLopt solver status: 1 ( NLOPT_SUCCESS: Generic success return value. )
#> 
#> Number of Iterations....: 17 
#> Termination conditions:  stopval: -Inf	xtol_rel: 1e-08	maxeval: 1000	ftol_rel: 0	ftol_abs: 0 
#> Number of inequality constraints:  0 
#> Number of equality constraints:    0 
#> Optimal value of objective function:  368.105912874334 
#> Optimal value of controls: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2.109093 4
#> 
#> 
## Optimal value of objective function:  368.105912874334
## Optimal value of controls: 2  ...  2  2.109093  4