  Abstract dynamic programming models are used to analyze $\lambda$-policy
iteration with randomization algorithms. Particularly, contractive models with
infinite policies are considered and it is shown that well-posedness of the
$\lambda$-operator plays a central role in the algorithm. The operator is known
to be well-posed for problems with finite states, but our analysis shows that
it is also well-defined for the contractive models with infinite states
studied. Similarly, the algorithm we analyze is known to converge for problems
with finite policies, but we identify the conditions required to guarantee
convergence with probability one when the policy space is infinite regardless
of the number of states. Guided by the analysis, we exemplify a data-driven
approximated implementation of the algorithm for estimation of optimal costs of
constrained linear and nonlinear control problems. Numerical results indicate
potentials of this method in practice.
