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

In this paper we present a framework to analyze the asymptotic behavior of two timescale stochastic approximation algorithms including those with set-valued mean fields. This paper builds on the works of Borkar and Perkins & Leslie. The framework presented herein is more general as compared to the synchronous two timescale framework of Perkins \& Leslie, however the assumptions involved are easily verifiable. As an application, we use this framework to analyze the two timescale stochastic approximation algorithm corresponding to the Lagrangian dual problem in optimization theory.