Abstract
Considering the progress in the maximum seeking method for multi-dimensional non-linear programing, recently the studies on the unimodal problem have developed rapidly and steadily to yield results. The multimodal problem, however, has remained unsolved in spite of its importance in the industry.
In this paper, a new method is proposed to contribute to solving the latter problem.
A noticeable feature of this method is to separate every peak which has a meaningful extremum from the wide n-dimensional surface using Monte Carlo integration skillfully. After the trial search is repeated a set number of times around the search points which are given by random arrangement, the several points whose values of the function are relatively high are selected and the spreading area (defined specially) of the peak is calculated. Whether to register it as an independent peak separated or not is decided on the basis of the position and the spreading area of every peak.
In this procedure, the criterion for the separation is given theoretically and also the necessary number of trials can be estimated by the theory which incorporates a statistical contemplation. In addition to theoretical consideration, a few experiments on the three kinds of test functions are made to confirm the performance separating the independent peaks. Experimental values agreed well with the thoretical results.
This method has such merits as being applicable to all functions without restriction, even discontinuous and to multi-n-dimensional problems even if n is considerably large. These are merits of the random method in general, and of course, this method has a weak point, in that the accuracy of results can not be given by anything except statistical values.
In order to reach a maximum point on the multimodal surface, the techniques in case of unimodal problems can be used after some independent peaks have been separated by this method.
