  This work develops new results for stochastic approximation algorithms. The
emphases are on treating algorithms and limits with discontinuities. The main
ingredients include the use of differential inclusions, set-valued analysis,
and non-smooth analysis, and stochastic differential inclusions. Under broad
conditions, it is shown that a suitably scaled sequence of the iterates has a
differential inclusion limit. In addition, it is shown for the first time that
a centered and scaled sequence of the iterates converges weakly to a stochastic
differential inclusion limit. The results are then used to treat several
application examples including Markov decision process, Lasso algorithms,
Pegasos algorithms, support vector machine classification, and learning. Some
numerical demonstrations are also provided.
