Abstract
Seeking the maximum point of an unknown multi-modal function is one of the most important problems for solving an optimal decision in the case where the criterion may have many local maxima. However, no useful method to find the global maximum point has been available yet.This paper presents a powerful method for multi-dimensional criterion functions. This is an extension of the method formerly developed for the one-dimensional case by authors.
The search procedure consists of three main steps: (i) to fit local models according to observed data, (ii) to essimate the location of the maximum point considering the peaks of the model and the uncertainty of the models due to lack of enough data, (iii) to search an area where the estimated maximum point exists.
Simulation studies on two 1-dimensional and four 2-dimensional test functions randomly selected provided fairly good results.
The correct maxima are detected with less than 0.5% errors. Number of observations in this method was almost the same as that of ramdom search in 1-dimensional case and about one-eighth times in 2-dimensional case.
