This thesis is concerned with the development and implementation
of a number of numerical algorithms for solving finite dimensional
nonlinear optimization problems. Most of the currently used
optimization methods use directly or indirectly a local quadratic
representation of the objective function.
The need to introduce non-quadratic models arises from the fact
that objective functions may not be represented adequately by quadratic
functions. Although complicated models can represent general nonlinear
functions more accurately than quadratics, they are more difficult to
deal with both analytically and numerically. Therefore there must be
a compromise between the generality of the model and the ease with which
it can be used in the context of optimization.
Various non-quadratic models have been tested in both gradient
and nongradient methods. Sophisticated gradient techniques such as
conjugate gradient and variable metric methods have been used with the
proposed models and compared with the traditional methods over a variety
of standard test functions.
One measure of the complexity of an optimization problem is its
size, measured in terms of the number of unknown variables. Despite
the fact that computation has been seriously dealt with and good algorithms
have been proposed in the past insufficient work has been done on very
large problems. Therefore most of the standard test problems were used
in their generalized form to gain insight into the efficiency of the use
of the non-quadratic models as the dimension of the problem increases.
The numerical results show that the use of non-quadratic models is
beneficial in most of the problems considered especially when the dimensionality
of the problem increases.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University