Approximate Max-Min Resource Sharing for Structured Concave Optimization

We present a Lagrangian decomposition algorithmwhich uses logarithmic potential reduction to compute an ε-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(ε+lnM)) iterations, a data independent bound which is optimal up to polylogarithmic factors for any fixed relative accuracy ε ∈ (0, 1). In the general structured case, B is the product of K convex blocks and each constraint function is block separable. For suchmodels, an iteration of our method requires a Θ(ε)-approximate solution of K independent block maximization problems which can be computed in parallel. AMS subject classification. 68Q25, 90C05, 90C27, 90C30, 90C06.