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

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.