Parallel Implementations Of The Nelder-Mead Simplex Algorithm For Unconstrained Optimization

We are interested in implementing direct search methods on parallel computers to solve the unconstrained minimization problem: Given a function f : IRn --? IR find an x E En that minimizes 1 (x). Our preliminary work has focused on the Nelder-Mead simplex algorithm. The origin of the algorithm can be found in a 1962 paper by Spendley, Hext and Himsworth;1 Nelder and Meade proposed an adaptive version which proved to be much more robust in practice. Dennis and Woods3 give a clear presentation of the standard Nelder-Mead simplex algorithm; Woods4 includes a more complete discussion of implementation details as well as some preliminary convergence results. Since descriptions of the standard Nelder-Mead simplex algorithm appear in Nelder and Mead,2 Dennis and Woods,3 and Woods,4 we will limit our introductory discussion to the advantages and disadvantages of the algorithm, as well as some of the features which make it so popular. We then outline the approaches we have taken and discuss our preliminary results. We conclude with a discussion of future research and some observations about our findings.