Distributed Min-max Optimization over Digraphs

This paper considers the problem of minimizing the maximum of several convex functions which are known by local nodes. First, the problem is transformed to a distributed constrained optimization. Then two methods, namely, constructing an exact penalty function and generating approximate projection, are proposed to handle constraints. Under a strongly connected unbalanced digraph, the two algorithms are both proved to converge to some common optimal solution, which is also validated by an example of localizing the Chebyshev center.