A New Maximum Searching Method for Unknown One-Dimensional Multimodal Functions

In this paper a global maximum searching method is developed for one-dimensional multimodal unknown functions. Justifiable properties of one-dimensional functions are analyzed, and they are used in the method of making approximate functions and global maximum searching. In maximum searching the domain is divided into a set of subdomains, and within the set, the sub-domain whose estimated maximum is the largest is chosen. At a properly selected point within the subdomain an evaluation of the function is made, and the sub-domain is dividbd into two smaller ones for more accurate estimation. This process is continued until the estimated maximum become equal to the evaluated maximum. For estimation of the maximum of a sub-domain a parameter of uncertainty is introduced. The larger the uncertainty is, the larger the estimated maximum is made. A large estimated maximum means either that the true local maximum is large or that a closer search of the subdomain is required to reduce the uncertainty. This uncertainty also plays a role of changing searching modes from global to concentrated local search and to checking if the maximum obtained is the true maximum. Performance of the method is evaluated for six test functions ranging from a unimodal up to a 13-modal one. The results show that the maximum points are obtained with high efficiency and with high reliability.