  We study two-stage stochastic optimization problems with random recourse,
where the adaptive decisions are multiplied with the uncertain parameters in
both the objective function and the constraints. To mitigate the computational
intractability of infinite-dimensional optimization, we propose a scalable
approximation scheme via piecewise linear and piecewise quadratic decision
rules. We then develop a data-driven distributionally robust framework with two
layers of robustness to address distributionally uncertainty. The emerging
optimization problem can be reformulated as an exact copositive program, which
admits tractable approximations in semidefinite programming. We design a
decomposition algorithm where smaller-size semidefinite programs can be solved
in parallel, which further reduces the runtime. Lastly, we establish the
performance guarantees of the proposed scheme and demonstrate its effectiveness
through numerical examples.
