tnewton | R Documentation |

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(),
...
)
```

`x0` |
starting point for searching the optimum. |

`fn` |
objective function that is to be minimized. |

`gr` |
gradient of function |

`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 |

`...` |
additional arguments passed to the function. |

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.

List with components:

`par` |
the optimal solution found so far. |

`value` |
the function value corresponding to |

`iter` |
number of (outer) iterations, see |

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

`message` |
character string produced by NLopt and giving additional information. |

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

Hans W. Borchers

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

`lbfgs`

```
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
```

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