Scaling rank-one updating formula and its application in unconstrained optimization

This thesis deals with algorithms used to solve unconstrained optimization problems. We analyse the properties of a scaling symmetric rank one (SSRl) update, prove the convergence of the matrices generated by SSRl to the true Hessian matrix and show that algorithm SSRl possesses the quadratic termination property with inexact line search. A new algorithm (OCSSRl) is presented, in which the scaling parameter in SSRl is choosen automatically by satisfying Davidon's criterion for an optimaly conditioned Hessian estimate. Numerical tests show that the new method compares favourably with BFGS. Using the OCSSRl update, we propose a hybrid QN algorithm which does not need to store any matrix. Numerical results show that it is a very promising method for solving large scale optimization problems. In addition, some popular technologies in unconstrained optimization are also discussed, for example, the trust region step, the descent direction with supermemory and. the detection of large residual in nonlinear least squares problems. The thesis consists of two parts. The first part gives a brief survey of unconstrained optimization. It contains four chapters, and introduces basic results on unconstrained optimization, some popular methods and their properties based on quadratic approximations to the objective function, some methods which are suitable for solving large scale optimization problems and some methods for solving nonlinear least squares problems. The second part outlines the new research results, and containes five chapters, In Chapter 5, the scaling rank one updating formula is analysed and studied. Chapter 6, Chapter 7 and Chapter 8 discuss the applications for the trust region method, large scale optimization problems and nonlinear least squares. A final chapter summarizes the problems used in numerical testing

Optimal Conditioning in the Convex Class of Rank Two Updates

Davidson's new quasi-Newton optimization algorithm selects the new inverse Hessian approximation H at each step to be the "optimally conditioned" member of a certain one-parameter class of rank two updates to the last inverse Hessian approximation H. In this paper, we show that virtually the same goals of conditioning can be achieved while restricting H to the convex class of updates. We therefore suggest that Davidson's algorithms using optimal conditioning, restrict the choice of H to members of the convex class