Adaptive Step Size for a Consensus based Distributed Subgradient Method in Generalized Mutual Assignment Problem

Generalized Mutual Assignment Problem (GMAP) is a multi-agent based distributed optimization where the agents try to obtain the most profitable job assignment. Since it is NP-hard and even a problem of judging the existence of a feasible solution is NP-complete, it is a challenging issue to solve GMAP. In this paper, a consensus based distributed subgradient method is considered to obtain feasible solutions of GMAP as good as possible. Adaptive step size which is calculated by the lower and estimated upper bounds is proposed for the step size in the subgradient method. In addition, a protocol how to estimate the upper bound is also proposed, where each agent do not have to synchronize it.