Stochastic recursive inclusion in two timescales with an application to the Lagrangian dual problem

A framework is presented to analyze the asymptotic behavior of two timescale stochastic approximation algorithms to include situations where the mean fields are set-valued. The framework is a natural generalization of the one developed by Borkar. Perkins and Leslie have developed a framework for asynchronous coupled stochastic approximation algorithms with set-valued mean fields. Our framework is however more general as compared to the synchronous version of the Perkins and Leslie framework.