Distributed Coupling Chance-Constraint Optimization under Unknown Uncertainty Distributions

In this paper, 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 the deterministic inequality constraints are related to the inverse cumulative density functions (ICDFs) of the unknown uncertainty distributions, a data-based approach is proposed for the approximations related to the ICDFs by the law of large numbers. For the sake of storing sampling data, a tree-based data structure is built, which is convenient to insert new data, expand the depth and width and query the corresponding approximations. Though the approximations of the ICDFs are monotonous, piecewise, but nonconvex globally, there exists locally convex property in the small enough neighborhoods of almost all 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 the 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.