Probabilistic Multimodal Optimization

Multimodal optimization, which aims to discover multiple satisfactory solutions simultaneously, has attracted increasing attention from researchers in the evolutionary computation community. With the aid of niching methods, evolutionary algorithms could simultaneously locate multiple satisfactory solutions in a single run. Although many multimodal evolutionary algorithms have been developed, they are confronted with two limitations: (1) in the niching stage, most niching-based multimodal methods need to compute pairwise Euclidean distances between individuals to separate the population into species, which gives rise to a high computational burden; and (2) in the optimization stage, most existing multimodal algorithms may have limitations in exploring the solution space, due to the utilization of traditional individual-based meta-heuristics, which may easily get trapped in local areas. To resolve the above issues, in this chapter, we introduce probabilistic multimodal optimization algorithms by presenting two probability-based frameworks for multimodal optimization. More specifically, we present a probability-based niching framework to accelerate the niching speed and a probability-based optimization framework to promote the optimization efficiency of multimodal algorithms, respectively. In the former, we utilize locality sensitive hashing to project individuals into buckets with probabilities to divide the population into species. Such a framework can be embedded into different niching methods to accelerate the niching speed. In the latter, we take advantage of the probability distribution of individuals to evolve the population along with a novel adaptive local search method. To instantiate these two frameworks, we customize them using two distinct approaches, respectively. More concretely, we embed the former into locally informed particle swarm optimization (LIPS) and neighborhood-based crowding differential evolution (NCDE), and customize the latter utilizing an explicit probability-based algorithm, the estimation of distribution algorithm (EDA), and an implicit probability-based algorithm, the continuous ant colony optimization algorithm (ACO), respectively. The efficiency and effectiveness of these two frameworks are carefully examined on a widely used multimodal benchmark set by means of comparing the associated customized algorithms with state-of-the-art multimodal methods. Lastly, the application of the proposed algorithms on multiple pedestrian detection problems is also presented. Experimentally, our approach is seen to perform competitively with traditional methods in this domain.