Extension of Stochastic Point Location for Multimodal Problems

Stochastic point location (SPL) involves a learning mechanism (LM) determining an optimal point on the line when the only inputs LM receives are stochastic information about the direction in which LM should move. The complexity of SPL comes from the stochastic responses of the environment, which may lead LM completely astray. SPL is a fundamental problem in optimization and was studied by many researchers during the last two decades, including improvement of its solution and all-pervasive applications. However, all existing SPL studies assume that the whole search space contains only one optimal point. Since a multimodal optimization problem (MMOP) contains multiple optimal solutions, it is significant to develop SPL's multimodal version. This article extends it from a unimodal problem to a multimodal one and proposes a parallel partition search (PPS) solution to address this issue. The heart of the proposed solution involves extracting the feature of the historical sampling information to distinguish the subintervals that contain the optimal points or not. Specifically, it divides the whole search space into multiple subintervals and samples them parallelly, then utilizes the feature of the historical sampling information to adjust the subintervals adaptively and to find the subintervals containing the optimal points. Finally, the optimal points are located within these subintervals according to their respective sampling statistics. The proof of the ϵ-optimal property for the proposed solution is presented. The numerical testing results demonstrate the power of the scheme.