  Distributed optimization is manifesting great potential in multiple fields,
e.g., machine learning, control, and resource allocation. Existing
decentralized optimization algorithms require sharing explicit state
information among the agents, which raises the risk of private information
leakage. To ensure privacy security, combining information security mechanisms,
such as differential privacy and homomorphic encryption, with traditional
decentralized optimization algorithms is a commonly used means. However, this
would either sacrifice optimization accuracy or incur heavy computational
burden. To overcome these shortcomings, we develop a novel privacy-preserving
decentralized optimization algorithm, called PPSD, that combines gradient
tracking with a state decomposition mechanism. Specifically, each agent
decomposes its state associated with the gradient into two substates. One
substate is used for interaction with neighboring agents, and the other
substate containing private information acts only on the first substate and
thus is entirely agnostic to other agents. For the strongly convex and smooth
objective functions, PPSD attains a $R$-linear convergence rate. Moreover, the
algorithm can preserve the agents' private information from being leaked to
honest-but-curious neighbors. Simulations further confirm the results.
