The use of non-quadratic models in optimization

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.