The corrective nonlinear simplex search algorithm and its application to the refinement problems of crystal structures

Abstract

This dissertation is concerned with the following unconstrained optimization problem: [See PFD for equation]. This problem is known as the refinement problem of crystal structures. Previous contributions of optimization methodology to the direct solution of crystal structures have made use of the phase relationships in the measured structure factors. This dissertation describes the application of a direct approach to the refinement problem of crystal structures using a nonlinear optimization methodology. The first part of this dissertation is concerned with the development of the corrective nonlinear simplex search technique. After investigating several nonlinear search techniques as applied to the refinement problem, we added a new feature to the nonlinear simplex algorithm which resulted in a more nearly optimal solution of the refinement problem. This algorithm, which is named the corrective nonlinear simplex search (CNSS) method, consists of two basic steps: 1. Search for the direction of each variable by an exploratory search method, 2. Then, search for the optimal positions by the flexible polygon search technique which requires four operations: reflection, expansion, contraction, and reduction..