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The method I present in this paper is suitable for smooth functions
where local minima exist. It is not a method for global optimization
where the global minimum is sought.
We define a local minimizer for and we
assume that and satisfy the ``standard
requirements'':
- 1.
*f* is twice differentiable,
- 2.
- ,
- 3.
- is positive definite
, i.e., for all (' denotes the adjoint).

where *N* is the dimension of the model vector and
the real space for the model vector .Any vector that satisfies the standard requirements is a local
minimizer of .
Newton's method is an iterative process where the
solution to the problem is updated as follows:

| |
(72) |

where is the updated solution at iteration
*k*+1, the step length computed by a linesearch
that ensures a sufficient decrease of and
the Hessian (or second derivative). In many circumstances the inverse
of the Hessian can't be computed directly. It happens for example
when the matrix is too big or when operators are used
rather than matrices.
Fortunately we might be able to compute an approximation of
the Hessian of . This strategy gives birth to quasi-Newton methods
where the way in which the Hessian is computed determines the method
Kelley (1999).
A possible update of the Hessian is given by the BFGS technique
Broyden (1969); Fletcher (1970); Goldfarb (1970); Shanno (1970).
The BFGS update is given by

| |
(73) |

where and
.In practice, however, we rather write the previous equation in terms of
the inverse matrices. We have then
| |
(74) |

In addition, we use the history of the iterations to compute the new
Hessian rather than a full storage of the matrix .This requires that a gradient step vector and a solution step
vector are kept in memory after each iteration.
Consequently this method might not been affordable with large data and
model space. In the next section
a modified version of the BFGS method that limits the storage needed
to compute the update of the Hessian is proposed.

** Next:** The limited memory BFGS
** Up:** Algorithm for minimizing the
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Stanford Exploration Project

5/5/2005