Distributed Coupling Chance-Constraint Optimization under Unknown Uncertainty Distributions

In this article, a distributed coupling chance-constraint optimization (3CO) problem under unknown uncertainty distributions is studied. An auxiliary variable is employed to assist decoupling and deterministic transformation, such that the coupling chance constraint is equivalent to a coupling equality constraint and a class of deterministic inequality constraints. Since deterministic inequality constraints are related to the inverse cumulative density functions (ICDFs) of unknown uncertainty distributions, a data-based approach is proposed for the approximations related to ICDFs by the law of large numbers. For the sake of storing sampling data, a tree-based data structure is built, in which it is convenient to insert new data, expand the depth and width, and query the corresponding approximations. Though the approximations of ICDFs are monotonous, piecewise, and globally nonconvex, there exists a locally convex property in the small enough neighborhoods of almost all the points. Thus, inspired by this perspective, a distributed optimization strategy is designed for the 3CO problem. By setting appropriate parameters, the stability and convergence of the strategy are guaranteed, and optimality is just influenced by the approximation ability of the tree-based data structure. Finally, some simulation results are provided to verify the effectiveness of the proposed strategy.