  This paper considers the distributed smooth optimization problem in which the
objective is to minimize a global cost function formed by a sum of local smooth
cost functions, by using local information exchange. The standard assumption
for proving exponential/linear convergence of first-order methods is the strong
convexity of the cost functions, which does not hold for many practical
applications. In this paper, we first show that the continuous-time distributed
primal-dual gradient algorithm converges to one global minimizer exponentially
under the assumption that the global cost function satisfies the restricted
secant inequality condition. This condition is weaker than the strong convexity
condition since it does not require convexity and the global minimizers are not
necessary to be unique. We then show that the discrete-time distributed
primal-dual algorithm constructed by using the Euler's approximation method
converges to one global minimizer linearly under the same condition. The
theoretical results are illustrated by numerical simulations.
