nlm {stats} R Documentation

## Non-Linear Minimization

### Description

This function carries out a minimization of the function `f` using a Newton-type algorithm. See the references for details.

### Usage

```nlm(f, p, hessian = FALSE, typsize=rep(1, length(p)), fscale=1,
print.level = 0, ndigit=12, gradtol = 1e-6,
stepmax = max(1000 * sqrt(sum((p/typsize)^2)), 1000),
steptol = 1e-6, iterlim = 100, check.analyticals = TRUE, ...)
```

### Arguments

 `f` the function to be minimized. If the function value has an attribute called `gradient` or both `gradient` and `hessian` attributes, these will be used in the calculation of updated parameter values. Otherwise, numerical derivatives are used. `deriv` returns a function with suitable `gradient` attribute. This should be a function of a vector of the length of `p` followed by any other arguments specified by the `...` argument. `p` starting parameter values for the minimization. `hessian` if `TRUE`, the hessian of `f` at the minimum is returned. `typsize` an estimate of the size of each parameter at the minimum. `fscale` an estimate of the size of `f` at the minimum. `print.level` this argument determines the level of printing which is done during the minimization process. The default value of `0` means that no printing occurs, a value of `1` means that initial and final details are printed and a value of 2 means that full tracing information is printed. `ndigit` the number of significant digits in the function `f`. `gradtol` a positive scalar giving the tolerance at which the scaled gradient is considered close enough to zero to terminate the algorithm. The scaled gradient is a measure of the relative change in `f` in each direction `p[i]` divided by the relative change in `p[i]`. `stepmax` a positive scalar which gives the maximum allowable scaled step length. `stepmax` is used to prevent steps which would cause the optimization function to overflow, to prevent the algorithm from leaving the area of interest in parameter space, or to detect divergence in the algorithm. `stepmax` would be chosen small enough to prevent the first two of these occurrences, but should be larger than any anticipated reasonable step. `steptol` A positive scalar providing the minimum allowable relative step length. `iterlim` a positive integer specifying the maximum number of iterations to be performed before the program is terminated. `check.analyticals` a logical scalar specifying whether the analytic gradients and Hessians, if they are supplied, should be checked against numerical derivatives at the initial parameter values. This can help detect incorrectly formulated gradients or Hessians. `...` additional arguments to `f`.

### Details

If a gradient or hessian is supplied but evaluates to the wrong mode or length, it will be ignored if `check.analyticals = TRUE` (the default) with a warning. The hessian is not even checked unless the gradient is present and passes the sanity checks.

From the three methods available in the original source, we always use method “1” which is line search.

### Value

A list containing the following components:

 `minimum` the value of the estimated minimum of `f`. `estimate` the point at which the minimum value of `f` is obtained. `gradient` the gradient at the estimated minimum of `f`. `hessian` the hessian at the estimated minimum of `f` (if requested). `code` an integer indicating why the optimization process terminated. 1:relative gradient is close to zero, current iterate is probably solution. 2:successive iterates within tolerance, current iterate is probably solution. 3:last global step failed to locate a point lower than `estimate`. Either `estimate` is an approximate local minimum of the function or `steptol` is too small. 4:iteration limit exceeded. 5:maximum step size `stepmax` exceeded five consecutive times. Either the function is unbounded below, becomes asymptotic to a finite value from above in some direction or `stepmax` is too small. `iterations` the number of iterations performed.

### References

Dennis, J. E. and Schnabel, R. B. (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, NJ.

Schnabel, R. B., Koontz, J. E. and Weiss, B. E. (1985) A modular system of algorithms for unconstrained minimization. ACM Trans. Math. Software, 11, 419–440.

`optim` and `nlminb`.

`constrOptim` for constrained optimization, `optimize` for one-dimensional minimization and `uniroot` for root finding. `deriv` to calculate analytical derivatives.

For nonlinear regression, `nls` may be better.

### Examples

```f <- function(x) sum((x-1:length(x))^2)
nlm(f, c(10,10))
nlm(f, c(10,10), print.level = 2)
str(nlm(f, c(5), hessian = TRUE))

f <- function(x, a) sum((x-a)^2)
nlm(f, c(10,10), a=c(3,5))
f <- function(x, a)
{
res <- sum((x-a)^2)