Distributed Finite-time Optimization for Compromise-Seeking Agents with Relative Preferences

In this paper we propose a distributed optimization problem with a global objective given by a weighted sum of local objectives, where each local weight encodes the absolute relevance of the local objective associated to an agent. In our settings, each agent is assumed to have only local (possibly inconsistent) information regarding the relative importance of its objective function with respect to its neighboring agents. Indeed, this allows to model scenarios where only partial knowledge is available to each agents, e.g., for privacy reasons. In this regard, we propose a distributed framework where agents cooperate to both negotiate their absolute relevance and solve the resulting optimization problem. The proposed framework ensures finite-time convergence under the assumption that for each local objective function the related Hessian matrix has eigenvalues that are lower-bounded by a known constant.