A particle swarm algorithm with broad applicability in shape-constrained estimation

In nonparametric function estimation, the inclusion of shape constraints can confer several advantages, including improved estimation accuracy, reduced sensitivity to smoothing parameters, and control over the qualitative appearance of the estimate. Finding shape-restricted estimates may require solving a difficult optimization problem, however, making these advantages hard to realize. A particle swarm algorithm is proposed to overcome this barrier and expand the possibilities for shape-constrained estimation. The algorithm uses a cooperative search strategy with two swarms, one focused on global exploration and one focused on local exploitation. The new heuristic has the added advantage of being a general tool, applicable without modification to a variety of estimators, constraints, and objective functions. The algorithm is demonstrated on a number of density estimation and regression problems, and the potential for further improvement is discussed. Supplementary materials, including source code, are available online.