  Privacy preservation is addressed for decentralized optimization, where $N$
agents cooperatively minimize the sum of $N$ convex functions private to these
individual agents. In most existing decentralized optimization approaches,
participating agents exchange and disclose states explicitly, which may not be
desirable when the states contain sensitive information of individual agents.
The problem is more acute when adversaries exist which try to steal information
from other participating agents. To address this issue, we propose a
privacy-preserving decentralized optimization approach based on ADMM and
partially homomorphic cryptography. To our knowledge, this is the first time
that cryptographic techniques are incorporated in a fully decentralized setting
to enable privacy preservation in decentralized optimization in the absence of
any third party or aggregator. To facilitate the incorporation of encryption in
a fully decentralized manner, we introduce a new ADMM which allows time-varying
penalty matrices and rigorously prove that it has a convergence rate of
$O(1/t)$. Numerical and experimental results confirm the effectiveness and low
computational complexity of the proposed approach.
