Fractional-Memory Distributed Online Optimization with Long-Term Constraints

This paper is concerned with distributed online optimization with long-term inequality constraints over time-varying unbalanced directed networks. A fractional-memory-based distributed online optimization algorithm is developed to exploit both current and historical first-order information. Its search direction is constructed using normalized power-law memory weights, whose distribution is regulated by the fractional order and whose memory horizon is determined by the maximum memory lag. An auxiliary-weight mechanism is incorporated to cope with the imbalance of directed communication, while the long-term constraint formulation allows projection onto a computationally tractable superset of the original feasible region, thereby avoiding costly constrained projections at every round. Under suitable assumptions and parameter choices, sublinear static regret and cumulative absolute constraint violation bounds are established. Simulations for a distributed online ridge regression problem demonstrate the feasibility of the proposed algorithm.