  This paper proposes distributed algorithms to solve robust convex
optimization (RCO) when the constraints are affected by nonlinear uncertainty.
We adopt a scenario approach by randomly sampling the uncertainty set. To
facilitate the computational task, instead of using a single centralized
processor to obtain a "global solution" of the scenario problem (SP), we resort
to {\it multiple interconnected processors} that are distributed among
different nodes of a network to simultaneously solve the SP. Then, we propose a
primal-dual sub-gradient algorithm and a random projection algorithm to
distributedly solve the SP over undirected and directed graphs, respectively.
Both algorithms are given in an explicit recursive form with simple iterations,
which are especially suited for processors with limited computational
capability. We show that, if the underlying graph is strongly connected, each
node asymptotically computes a common optimal solution to the SP with a
convergence rate $O(1/(\sum_{t=1}^k\zeta^t))$ where $\{\zeta^t\}$ is a sequence
of appropriately decreasing stepsizes. That is, the RCO is effectively solved
in a distributed way. The relations with the existing literature on robust
convex programs are thoroughly discussed and an example of robust system
identification is included to validate the effectiveness of our distributed
algorithms.
