1 research outputs found
Adaptive Sequential SAA for Solving Two-stage Stochastic Linear Programs
We present adaptive sequential SAA (sample average approximation) algorithms
to solve large-scale two-stage stochastic linear programs. The iterative
algorithm framework we propose is organized into \emph{outer} and \emph{inner}
iterations as follows: during each outer iteration, a sample-path problem is
implicitly generated using a sample of observations or ``scenarios," and solved
only \emph{imprecisely}, to within a tolerance that is chosen
\emph{adaptively}, by balancing the estimated statistical error against
solution error. The solutions from prior iterations serve as \emph{warm starts}
to aid efficient solution of the (piecewise linear convex) sample-path
optimization problems generated on subsequent iterations. The generated
scenarios can be independent and identically distributed (iid), or dependent,
as in Monte Carlo generation using Latin-hypercube sampling, antithetic
variates, or randomized quasi-Monte Carlo. We first characterize the
almost-sure convergence (and convergence in mean) of the optimality gap and the
distance of the generated stochastic iterates to the true solution set. We then
characterize the corresponding iteration complexity and work complexity rates
as a function of the sample size schedule, demonstrating that the best
achievable work complexity rate is Monte Carlo canonical and analogous to the
generic optimal complexity for non-smooth convex
optimization. We report extensive numerical tests that indicate favorable
performance, due primarily to the use of a sequential framework with an optimal
sample size schedule, and the use of warm starts. The proposed algorithm can be
stopped in finite-time to return a solution endowed with a probabilistic
guarantee on quality