Iterated Grid Search Algorithm on Unimodal Criteria

by

Jinhyo Kim

PhD Dissertation submitted to the Faculty of the Virginia Tech in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Statistics

Approved

Krutchkoff and Terrell

June 2, 1997
Blacksburg, Virginia

Abstract

The unimodality of a function seems a simple concept. But in the Euclidean space R^m, m=3,4,..., it is not easy to define. We have an easy tool to find the minimum point of a unimodal function. The goal of this project is to formalize and support distinctive strategies that typically guarantee convergence. Support is given both by analytic arguments and simulation study. Application is envisioned in low-dimensional but non-trivial problems. The convergence of the proposed iterated grid search algorithm is presented along with the results of particular application studies. It has been recognized that the derivative methods, such as the Newton-type method, are not entirely satisfactory, so a variety of other tools are being considered as alternatives. Many other tools have been rejected because of apparent manipulative difficulties. But in our current research, we focus on the simple algorithm and the guaranteed convergence for unimodal function to avoid the possible chaotic behavior of the function. Furthermore, in case the loss function to be optimized is not unimodal, we suggest a weaker condition: almost (noisy) unimodality, under which the iterated grid search finds an estimated optimum point. Subject Classification: statistical computing, nonlinear estimation, statistical optimization, statistical simulation Key Words: Iterated Grid Search, grid, dichotomous search, unimodality, quasi-convexity, envelope, condition number, derivative-free

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