# tnewton

##### 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.

##### 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:

the optimal solution found so far.

the function value corresponding to `par`

.

number of (outer) iterations, see `maxeval`

.

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

character string produced by NLopt and giving additional information.

##### 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).

##### See Also

##### Examples

```
# NOT RUN {
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
# }
```

*Documentation reproduced from package nloptr, version 1.2.1, License: LGPL-3*