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

Exponential Convergence for Distributed Smooth Optimization Under the Restricted Secant Inequality Condition