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In this paper, we study the problem of consensus-based distributedoptimization where a network of agents, abstracted as a directed graph, aims tominimize the sum of all agents' cost functions collaboratively. In existingdistributed optimization approaches (Push-Pull/AB) for directed graphs, allagents exchange their states with neighbors to achieve the optimal solutionwith a constant stepsize, which may lead to the disclosure of sensitive andprivate information. For privacy preservation, we propose a novelstate-decomposition based gradient tracking approach (SD-Push-Pull) fordistributed optimzation over directed networks that preserves differentialprivacy, which is a strong notion that protects agents' privacy against anadversary with arbitrary auxiliary information. The main idea of the proposedapproach is to decompose the gradient state of each agent into two sub-states.Only one substate is exchanged by the agent with its neighbours over time, andthe other one is kept private. That is to say, only one substate is visible toan adversary, protecting the privacy from being leaked. It is proved that undercertain decomposition principles, a bound for the sub-optimality of theproposed algorithm can be derived and the differential privacy is achievedsimultaneously. Moreover, the trade-off between differential privacy and theoptimization accuracy is also characterized. Finally, a numerical simulation isprovided to illustrate the effectiveness of the proposed approach.

A Differential Private Method for Distributed Optimization in Directed Networks via State Decomposition