Distributed Algorithms for Robust Convex Optimization via the Scenario Approach

TL;DR

This paper introduces distributed algorithms for solving robust convex optimization problems affected by nonlinear uncertainties, using a scenario approach and multiple interconnected processors to achieve convergence.

Contribution

It develops primal-dual and random projection algorithms for distributed robust convex optimization over networks, with proven convergence and applicability to systems with limited computational resources.

Findings

Algorithms converge to a common optimal solution under strongly connected graphs.

Distributed approach effectively solves robust convex optimization problems.

Validated with an example in robust system identification.

Abstract

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.