Descent With Approximate Multipliers is Enough: Generalising Max-Weight

We study the use of approximate Lagrange multipliers and discrete actions in solving convex optimisation problems. We observe that descent, which can be ensured using a wide range of approaches (gradient, subgradient, Newton, etc.), is orthogonal to the choice of multipliers. Using the Skorokhod representation for a queueing process we show that approximate multipliers can be constructed in a number of ways. These observations lead to the generalisation of (i) essentially any descent method to encompass use of discrete actions and queues and (ii) max-weight scheduling to encompass new descent methods including those with unsynchronised updates such as block coordinate descent. This also allows consideration of communication delays and of updates at varying time-scales within the same clean and consistent framework.