# Non-linear minimization routine

### Description

This function performs efficiently the optimization of the likelihood functions for multinomial logit models

### Usage

1 2 3 |

### Arguments

`logLik` |
the likelihood function to be maximized, |

`start` |
the initial value of the vector of coefficients, |

`method` |
the method used, one of |

`iterlim` |
the maximum number of iterations, |

`tol` |
the value of the criteria for the gradient, |

`ftol` |
the value of the criteria for the function, |

`steptol` |
the value of the criteria for the step, |

`print.level` |
one of (0, 1, 2), the details of the printing
messages. If |

`constPar` |
a numeric or a character vector which indicates that some parameters should be treated as constant, |

`...` |
further arguments passed to |

### Details

The optimization is performed by updating, at each iteration, the vector of parameters by the amount step * direction, where step is a positive scalar and direction = H^-1 * g, where g is the gradient and H^-1 is an estimation of the inverse of the hessian. The choice of H^-1 depends on the method chosen :

if `method='nr'`

, H is the hessian (i.e. is the second derivates
matrix of the likelihood function),

if `method = 'bhhh'`

, H is the outer-product of the individual
contributions of each individual to the gradient,

if `method = 'bfgs'`

, H^-1 is updated at each iteration using a
formula that uses the variations of the vector of parameters and the
gradient. The initial value of the matrix is the inverse of the
outer-product of the gradient (i.e. the bhh estimator of the
hessian).

The initial step is 1 and, if the new value of the function is less than the previous value, it is divided by two, until a higher value is obtained.

The routine stops when the gradient is sufficiently close to 0. The
criteria is g * H^-1 * g which is compared to the `tol`

argument. It also may stops if the number of iterations equals
`iterlim`

.

The function `f`

has a `initial.value`

argument which is the
initial value of the likelihood. The function is then evaluated a
first time with a step equals to one. If the value is lower than the
initial value, the step is divided by two until the likelihood
increases. The gradient is then computed and the function returns as
attributes the gradient is the step. This method is more efficient
than other functions available for R :

For the `optim`

and the `maxLik`

functions, the function and
the gradient should be provided as separate functions. But, for
multinomial logit models, both depends on the probabilities which are
the most time-consuming elements of the model to compute.

For the `nlm`

function, the fonction returns the gradient as an
attribute. The gradient is therefore computed at each iteration, even
when the function is computed with a step that is unable to increase
the value of the likelihood.

Previous versions of `mlogit`

depended on the `'maxLik'`

package. We kept the same interface, namely the `start`

,
`method`

, `iterlim`

, `tol`

, `print.level`

and
`constPar`

arguments.

The default method is `'bfgs'`

, which is known to perform well,
even if the likelihood function is not well behaved and the default
value for `print.level=1`

, which means moderate printing.

A special default behavior is performed if a simple multinomial logit
model is estimated. Indeed, for this model, the likelihood function is
concave, the analytical hessian is simple to write and the
optimization is straightforward. Therefore, in this case, the default
method is `'nr'`

and `print.level=0`

.

### Value

a list that contains the followings elements :

`optimum` |
the value of the function at the optimum, with
attributes: |

`coefficients` |
the vector of the parameters at the optimum, |

`est.stat` |
a list that contains some information about the
optimization : |

### Author(s)

Yves Croissant