Networked Parallel Algorithms for Robust Convex Optimization via the Scenario Approach

Abstract

This paper proposes a parallel computing framework to distributedly solve robust convex optimization (RCO) when the constraints are affected by nonlinear uncertainty. To this end, 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 parallel processors} that are distributed among different nodes of a network. 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 parallel 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 networked parallel algorithms.