  In this paper, we design, analyze, and implement a variant of the two-loop
L-shaped algorithms for solving two-stage stochastic programming problems that
arise from important application areas including revenue management and power
systems. We consider the setting in which it is intractable to compute exact
objective function and (sub)gradient information, and instead, only estimates
of objective function and (sub)gradient values are available. Under common
assumptions including fixed recourse and bounded (sub)gradients, the algorithm
generates a sequence of iterates that converge to a neighborhood of optimality,
where the radius of the convergence neighborhood depends on the level of the
inexactness of objective function estimates. The number of outer and inner
iterations needed to find an approximate optimal iterate is provided. Finally,
we show a sample complexity result for the algorithm with a Polyak-type
step-size policy that can be extended to analyze other situations. We also
present a numerical study that verifies our theoretical results and
demonstrates the superior empirical performance of our proposed algorithms over
classic solvers.
