4 research outputs found
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
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An evaluation of cluster analysis and related multivariate techniques for operational research
The following work. is an investigation into methods of Cluster Analysis and Ordination. The main objective of this thesis has been to investigate the capabilities of these methods for practical usage. An important subsidiary aim has been to collect together related work which has been carried out in many different areas - ecology, biology, archaeology, psychology, etc.., into, one work. After a_ brief introduction to the general concepts of multivariate analysis in Section A of the thesis, Section B gives an introductory account of the methods of Clustering, Ordination and Seriation, putting them into the context of the by now better established multivariate techniques. Section C considers Cluster Analysis in depth, explaining and examining various methods reported in the literature, together with methods developed by the author. The suitability of the methods for practical use is discussed and, decision rules are set out for the choice of method to be used in any particular study, based on the results of extensive comparative tests of the methods. In Section D the various ordination methods are. considered, giving an, overall viere and relating the methods to each other. Particular emphasis is paid to the rather neglected metric methods. Section E, after a survey of published applications of the methods, suggests new areas where the methods previously discussed could be valuable aids for data investigation and problem solving. An Addenda is included which describes several operational research case studies using these methods. Computer programs are given for the most successful of the newly introduced cluster methods, and an extensive reference section is also included