Approximate max-min resource sharing for structured concave optimization

We present a Lagrangian decomposition algorithm which uses logarithmic potential reduction to compute an epsilon -approximate solution of the general max-min resource sharing problem with M nonnegative concave constraints on a convex set B. We show that this algorithm runs in O(M (epsilon (-2) + In M)) iterations, a data independent bound which is optimal up to polylogarithmic factors for any fixed relative accuracy epsilon is an element of (0,1). In the block-angular case, B is the product of K convex sets ( blocks) and each constraint function is block separable. For such models, an iteration of our method requires a Theta (epsilon) approximate solution of K independent block maximization problems which can be computed in parallel.