Distributed Optimal Power Dispatch of On/Off Devices via Newton-like Neural Network Dynamics

This paper aims to solve a distributed optimization problem which naturally occurs in optimal dispatch of generators and flexible load devices that are limited to binary operational states, e.g. on and off. This problem is formulated as a binary program with a cost function defined by the summation of agent costs plus a global mismatch/penalty term. We propose a modification of the Hopfield Neural Network (HNN) dynamics in order to solve this problem while incorporating a novel Newton-like weighting factor. This addition lends itself to fast avoidance of saddle points, which the gradient-like HNN is susceptible to. Turning to a multi-agent setting, we reformulate the problem and develop a distributed implementation of the Newton-like dynamics. We show that if a local solution to the distributed reformulation is obtained, it is also a local solution to the centralized problem. A main contribution of this work is to show that the probability of converging to a saddle point of an appropriately defined energy function in both the centralized and distributed settings is zero under light assumptions. Finally, we enlarge our algorithm with an annealing technique which gradually "learns" a feasible binary solution. Simulation results demonstrate that the proposed methods are competitive with centralized greedy and SDP relaxation approaches in terms of solution quality, while the main advantage of our approach is a significant improvement in runtime over the SDP relaxation method and the distributed quality of implementation.