Distributed Adaptive Algorithm for Resource Allocation Problem Over Weight-Unbalanced Graphs

The paper presents the theoretical results on the resource allocation problems with local feasible convex set constraints under strongly connected graphs, where the objective function is non-smooth. It is worth noting that the communication graphs are general and include the weight-unbalanced directed graphs. Based on projection method, a distributed continuous-time adaptive algorithm is designed to achieve the optimal allocation. Moreover, the finite-time projection scheme is adopted to estimate a positive right-eigenvector of the out-Laplacian matrix from any initial value, which is a good indication for the robustness of the distributed algorithm when the communication topology is attacked and changed at some point. By convex and non-smooth analysis, we proved that the output variable asymptotically converges to the optimal allocation. Besides, the exponential convergence of the algorithm is analyzed without considering the local convex constraints. Finally, two illustrative examples are performed to substantiate the theoretical results under weight-unbalanced communication graphs.