newton | R Documentation |

Generalized newton optimizer used for the inner optimization problem.

```
newton(
par,
fn,
gr,
he,
trace = 1,
maxit = 100,
tol = 1e-08,
alpha = 1,
smartsearch = TRUE,
mgcmax = 1e+60,
super = TRUE,
silent = TRUE,
ustep = 1,
power = 0.5,
u0 = 1e-04,
grad.tol = tol,
step.tol = tol,
tol10 = 0.001,
env = environment(),
...
)
```

`par` |
Initial parameter. |

`fn` |
Objective function. |

`gr` |
Gradient function. |

`he` |
Sparse hessian function. |

`trace` |
Print tracing information? |

`maxit` |
Maximum number of iterations. |

`tol` |
Convergence tolerance. |

`alpha` |
Newton stepsize in the fixed stepsize case. |

`smartsearch` |
Turn on adaptive stepsize algorithm for non-convex problems? |

`mgcmax` |
Refuse to optimize if the maximum gradient component is too steep. |

`super` |
Supernodal Cholesky? |

`silent` |
Be silent? |

`ustep` |
Adaptive stepsize initial guess between 0 and 1. |

`power` |
Parameter controlling adaptive stepsize. |

`u0` |
Parameter controlling adaptive stepsize. |

`grad.tol` |
Gradient convergence tolerance. |

`step.tol` |
Stepsize convergence tolerance. |

`tol10` |
Try to exit if last 10 iterations not improved more than this. |

`env` |
Environment for cached Cholesky factor. |

`...` |
Currently unused. |

If `smartsearch=FALSE`

this function performs an ordinary newton optimization
on the function `fn`

using an exact sparse hessian function.
A fixed stepsize may be controlled by `alpha`

so that the iterations are
given by:

`u_{n+1} = u_n - \alpha f''(u_n)^{-1}f'(u_n)`

If `smartsearch=TRUE`

the hessian is allowed to become negative definite
preventing ordinary newton iterations. In this situation the newton iterations are performed on
a modified objective function defined by adding a quadratic penalty around the expansion point `u_0`

:

`f_{t}(u) = f(u) + \frac{t}{2} \|u-u_0\|^2`

This function's hessian ( `f''(u)+t I`

) is positive definite for `t`

sufficiently
large. The value `t`

is updated at every iteration: If the hessian is positive definite `t`

is
decreased, otherwise increased. Detailed control of the update process can be obtained with the
arguments `ustep`

, `power`

and `u0`

.

List with solution similar to `optim`

output.

`newtonOption`

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