  This paper addresses the problem of distributed optimization, where a network
of agents represented as a directed graph (digraph) aims to collaboratively
minimize the sum of their individual cost functions. Existing approaches for
distributed optimization over digraphs, such as Push-Pull, require agents to
exchange explicit state values with their neighbors in order to reach an
optimal solution. However, this can result in the disclosure of sensitive and
private information. To overcome this issue, we propose a
state-decomposition-based privacy-preserving finite-time push-sum (PrFTPS)
algorithm without any global information, such as network size or graph
diameter. Then, based on PrFTPS, we design a gradient descent algorithm
(PrFTPS-GD) to solve the distributed optimization problem. It is proved that
under PrFTPS-GD, the privacy of each agent is preserved and the linear
convergence rate related to the optimization iteration number is achieved.
Finally, numerical simulations are provided to illustrate the effectiveness of
the proposed approach.
