Distributed MIN-MAX Optimization Application to Time-optimal Consensus: An Alternating Projection Approach

TL;DR

This paper introduces an alternating projection algorithm for distributed MIN-MAX convex optimization, effectively solving time-optimal consensus problems in multi-agent systems through iterative projections and simulations.

Contribution

It presents a novel algorithm combining Bregman's and Dykstra's methods to solve distributed MIN-MAX problems, applied to multi-agent time-optimal consensus.

Findings

Algorithm successfully finds time-optimal consensus states.

Simulations confirm the effectiveness and convergence of the method.

Applicable to various multi-agent coordination scenarios.

Abstract

In this paper, we proposed an alternating projection based algorithm to solve a class of distributed MIN-MAX convex optimization problems. We firstly transform this MINMAX problem into the problem of searching for the minimum distance between some hyper-plane and the intersection of the epigraphs of convex functions. The Bregman's alternating method is employed in our algorithm to achieve the distance by iteratively projecting onto the hyper-plane and the intersection. The projection onto the intersection is obtained by cyclic Dykstra's projection method. We further apply our algorithm to the minimum time multi-agent consensus problem. The attainable states set for the agent can be transformed into the epigraph of some convex functions, and the search for time-optimal state for consensus satisfies the MIN-MAX problem formulation. Finally, the numerous simulation proves the validity of our algorithm.