module Bfgs:BFGS (Broyden-Fletcher-Goldfarb-Shanno) quasi-Newton local optimization method.`sig`

..`end`

The pure Newton method uses a local quadratic approximation of the
objective function `f`

being minimized:
`f(x)=f(x0)+ t(x-x0)G + t(x-x0)H(x-x0)`

where `G`

and `H`

are respectively the gradient vector and the hessian
matrix evaluated at `x0`

, and `t`

is the transpose operator.
When this approximation is valid, and when the function is locally convex,
the minimum is found at `x`

when `G + H(x-x0) = 0`

. This gives us the
Newton step `x= x0 - inv(H)G`

, where `inv(H)`

is the inverse hessian.

Quasi-Newton methods rely on an iterative approximation of the inverse
hessian, usually starting with the identity matrix as an initial
approximation (which means that we follow the gradient descent direction).
At each step `k`

of the algorithm, the function is minimized along the
search direction.
Let us call `lambda`

the step size along the quasi-Newton direction,
and `h(k)`

the approximation of the INVERSE hessian at point `x(k)`

.

The algorithm is the following. At each step `k`

:

1) minimize the function along the quasi-Newton direction. The line search
method minimizes `f(x(k)-lambda.h(k)G)`

as a function of `lambda`

.

2) make a step in the quasi-Newton direction. If the minimum along
that direction is found at `lambda_k`

, the quasi-Newton step is then
`x(k+1)= x(k) - lambda_k.h(k)G`

3) compute a new approximation `h(k+1)`

of the inverse hessian.
Several expressions can be found for this update. We used the
analytical expression:
`h(k+1)= t(I-yk.t(sk)/t(yk).sk).h(k).(I-yk.t(sk)/t(yk).sk) - sk.t(sk)/t(yk).sk`

where `yk= G(k+1) - G(k)`

and `sk= x(k+1) - x(k)`

The search stops when the value of the objective function falls beyond and absolute tolerance, or when it cannot be improved anymore.

Now that we have presented the main features of the algorithm, let us talk about a few additional subtleties. Three line search methods have been implemented: backracking, golden section search, and Brent's parabolic interpolation.

The backtracking starts with an initial step (default size is 1) along
the quasi-Newton direction.

`val verbose : ``bool ref`

Verbose printing on standard output.

`val logpoints : ``string ref`

Filename used to log the points explored at each step of the
BFGS search.

`type `

linesearch_method =

`|` |
`Backtracking` |
`(*` | ```
RECOMMENDED. Line search with a backtracking method, starting
with an initial step (default 1) in a descent direction, and
reducing the step until the objective function is sufficiently
improved (Armijo-Goldstein condition to ensure convergence).
``` | `*)` |

`|` |
`Golden` |
`(*` | `Line search using golden section method` | `*)` |

`|` |
`Brent` |

`val linesearch_method : ``linesearch_method ref`

Line search along the quasi-Newton direction.
Currently, there are three possible methods: backtracking (recommended),
golden section search, Brent's parabolic interpolation.

`bt_acctol`

, `bt_reduc`

, and `bt_epsilon`

are parameters for the
backtracking method. Backtracking constists in trying a candidate step
along the search direction. If the objective function is sufficiently
decreased, then the step is accepted. The decrease must be at least equal
to `!bt_acctol`

multiplied by the norm of the gradient projection along the
search direction (simplified Armijo-Goldstein condition).
Otherwise, if the current step is not accepted, a new step is tried
by multiplying the current step by a reduction factor `!bt_reduc`

.
The search stops when the step size falls under `!bt_epsilon`

.

`ls_abstol`

, `ls_reltol`

, and `ls_max_iter`

are
parameters for golden section search and Brent's parabolic interpolation.
For these methods, the search stops when the range of the search interval
falls under an absolute tolerance `!ls_abstol`

, or when it cannot be
significantly reduced (relative tolerance `!ls_reltol`

). If none of these
criteria is met, the search stops after `!ls_max_iter`

iterations.

`val bt_acctol : ``float ref`

Tolerance for accepting the bactracking step.

`val bt_reduc : ``float ref`

Backtracking step size reduction factor.

`val bt_epsilon : ``float ref`

Backtracking stops when steps size along the
search direction falls under

`!bt_epsilon`

.`val ls_max_iter : ``int ref`

Maximum number of iterations for the line search (Golden section
search or Brent's method).

`val ls_abstol : ``float ref`

Absolute tolerance for the stopping criterion (Golden section
search or Brent's method).

`val ls_reltol : ``float ref`

Relative tolerance for the stopping criterion (Golden section
search or Brent's method).

`val step : ``float ref`

Initial step length for the line search

`val bfgs : ``(float array -> float) ->`

(float array -> float array) ->

float array ->

int ->

float ->

float -> int -> (int -> bool) -> int * float array * float * float array

`bfgs f g xinit max_iter abstol reltol freq_verbose debug`

returns a tuple `(i,x,gx,hx)`

where `i`

is the iteration at which the
BFGS search stopped, `x`

is the best local minimum found for the function
`f`

, `gx`

is the value of the gradient `g`

at this point, and `hx`

is
the approximation of the inverse hessian at the minimum.
The search of a minimum stops when the maximum number of iterations
is reached or when two successive values `f1`

and `f2`

of the objective
function are closer than the absolute tolerance `abstol`

, or when the
relative difference between the two values falls under the relative
tolerance `reltol`

.

The exact formulation of the two last conditions is:
`|f2-f1|< abstol`

or `|f2-f1|< reltol*(|f1|+reltol)`

.
We add `reltol`

to `|f1|`

to take account of the case when
`f1`

is zero).

`xinit`

is the initial point at which the search starts,
`freq_verbose`

is the frequency at which some information is printed
on the standard output.
`debug`

is a function of the iteration number `i`

. Outputs for debugging
are printed when `debug i`

is true.