A conjugate gradient quasi-newton method for structural optimisation

Structural optimisation involves the constrained minimisation of functions or functionals and the majority of the methods employed utilise in some manner algorithms developed for unconstrained optimisation. Two unconstrained optimisation methods of historical and current significance are the conjugate gradient method and the quasi-Newton method. The conjugate gradient method in particular remains a popular technique in optimisation practice because of its simplicity and rapid convergence for well-conditioned problems. Both the conjugate gradient and quasi-Newton method offer quadratic termination, i.e. for exact arithmetic and an n-dimensional problem both methods will converge in a maximum of n steps. This paper is concerned with assessing the performance of a combined conjugate gradient and quasi-Newton method for a number of relatively simple non-linear structural optimisation problems. The problems considered are designed to be poorly conditioned as the new method is shown to be particularly adept at solving these types of problems. The features of the new method are quadratic termination and that matrix updating is present, similar to the quasi-Newton method. However, this latter feature is optional and if no updating takes place the method reduces to the standard conjugate gradient method whilst if full updating is performed the quasi-Newton method is obtained. It is shown in the paper that best performance is obtained with partial updating and the extent depends on the ill-conditioning of the problem.