Distributed robust optimization for multi-agent systems with guaranteed finite-time convergence

TL;DR

This paper introduces a distributed algorithm for multi-agent systems that guarantees finite-time convergence to a near-optimal feasible consensus solution for robust convex optimization problems with uncertainty.

Contribution

It develops a novel distributed approach with lower and upper bounding procedures, and termination methods, ensuring finite-time convergence with guaranteed accuracy.

Findings

The algorithm achieves finite-time convergence to a near-optimal solution.

It effectively handles bounded uncertainty in distributed convex optimization.

Numerical results demonstrate the algorithm's effectiveness and convergence properties.

Abstract

A novel distributed algorithm is proposed for finite-time converging to a feasible consensus solution satisfying global optimality to a certain accuracy of the distributed robust convex optimization problem (DRCO) subject to bounded uncertainty under a uniformly strongly connected network. Firstly, a distributed lower bounding procedure is developed, which is based on an outer iterative approximation of the DRCO through the discretization of the compact uncertainty set into a finite number of points. Secondly, a distributed upper bounding procedure is proposed, which is based on iteratively approximating the DRCO by restricting the constraints right-hand side with a proper positive parameter and enforcing the compact uncertainty set at finitely many points. The lower and upper bounds of the global optimal objective for the DRCO are obtained from these two procedures. Thirdly, two distributed termination methods are proposed to make all agents stop updating simultaneously by exploring whether the gap between the upper and the lower bounds reaches a certain accuracy. Fourthly, it is proved that all the agents finite-time converge to a feasible consensus solution that satisfies global optimality within a certain accuracy. Finally, a numerical case study is included to illustrate the effectiveness of the distributed algorithm.