  An effective numerical method is presented for optimizing model parameters
that can be applied to any type of system of non-linear equations and any
number of data-points, which does not require explicit formulation of the
objective function or its partial derivatives. The numerics are reduced to
solving a non-linear least squares problem, which uses the Levenberg-Marquardt
algorithm and the Jacobian is approximated by applying rank-one updates using
Broyden's method. An advantage of this methodology over conventional approaches
is that the partial derivatives of the objective function do not have to be
analytically calculated. For instance, there may be situations where one cannot
formulate the partial derivatives, such as cases involving an objective
function that itself contains a nested optimization problem. Moreover, a line
search algorithm is also described that ensures that the Armijo conditions are
satisfied and that convergence is assured, which makes the success of the
approach insensitive to the initial estimates of the model parameters. The
foregoing numerical methods are described with respect to the development of
the Optima software to solve inverse problems, which are reduced to non-linear
least squares problems. This computational approach has proven to be
particularly useful at solving inverse problems of very complex physical models
that cannot be optimized directly in a practical way.

Numerical Analysis;Materials Science;Numerical Analysis