Fully distributed resource allocation over unbalanced digraphs in prescribed time: A relaxed time-base generator approach

This paper focuses on three critical aspects of designing distributed optimization algorithms in real-world scenarios: feasibility, convergence time, and applicability to unbalanced networks. A specific class of resource allocation problems (RAPs) are addressed with these challenges in mind. These RAPs occur over unbalanced digraphs, have strict time constraints, and require continuous resource-demand balance. To tackle these challenges, a relaxed framework of time-base generators (TBGs) is presented. Sufficient conditions are then derived to guarantee that TBGs achieve prescribed-time convergence for distributed optimization. Secondly, leveraging the designed TBGs, a new prescribed-time distributed continuous-time optimization algorithm specifically suited for RAPs over unbalanced digraphs is firstly developed. This algorithm excels at solving these problems within user-defined and arbitrary timeframes. Thirdly, a fully distributed design of the above prescribed-time optimization algorithm is put forward to eliminate the stringent global topology knowledge. It is formally proved that both algorithms achieve the optimal resource allocation within prescribed-time convergence. Furthermore, both algorithms ensure continuous feasibility by maintaining resource-demand balance throughout operation and offer a significant advantage in simplicity compared to existing primal–dual methods and their variants. The proposed algorithms exclude the need for Lagrangian multipliers in the design process, leading to a more straightforward and convenient approach to handling the coupled resource-demand constraint. Finally, the effectiveness of the proposed algorithms is substantiated through numerical simulations.