  Two-stage stochastic programming is a popular framework for optimization
under uncertainty, where decision variables are split between first-stage
decisions, and second-stage (or recourse) decisions, with the latter being
adjusted after uncertainty is realized. These problems are often formulated
using Sample Average Approximation (SAA), where uncertainty is modeled as a
finite set of scenarios, resulting in a large "monolithic" problem, i.e., where
the model is repeated for each scenario. The resulting models can be
challenging to solve, and several problem-specific decomposition approaches
have been proposed. An alternative approach is to approximate the expected
second-stage objective value using a surrogate model, which can then be
embedded in the first-stage problem to produce good heuristic solutions. In
this work, we propose to instead model the distribution of the second-stage
objective, specifically using a quantile neural network. Embedding this
distributional approximation enables capturing uncertainty and is not limited
to expected-value optimization, e.g., the proposed approach enables
optimization of the Conditional Value at Risk (CVaR). We discuss optimization
formulations for embedding the quantile neural network and demonstrate the
effectiveness of the proposed framework using several computational case
studies including a set of mixed-integer optimization problems.
