Arunselvan Ramaswamy

A Generalization of the Borkar-Meyn Theorem for Stochastic Recursive Inclusions

In this paper the stability theorem of Borkar and Meyn is extended to include the case when the mean field is a differential inclusion. Two different sets of sufficient conditions are presented that guarantee the stability and convergence of stochastic recursive inclusions. Our work builds on the works of Benaim, Hofbauer and Sorin as well as Borkar and Meyn. As a corollary to one of the main theorems, a natural generalization of the Borkar and Meyn Theorem follows. In addition, the original theorem of Borkar and Meyn is shown to hold under slightly relaxed assumptions. Finally, as an application to one of the main theorems we discuss a solution to the approximate drift problem.

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.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.