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We study two-stage stochastic optimization problems with random recourse,where the adaptive decisions are multiplied with the uncertain parameters inboth the objective function and the constraints. To mitigate the computationalintractability of infinite-dimensional optimization, we propose a scalableapproximation scheme via piecewise linear and piecewise quadratic decisionrules. We then develop a data-driven distributionally robust framework with twolayers of robustness to address distributionally uncertainty. The emergingoptimization problem can be reformulated as an exact copositive program, whichadmits tractable approximations in semidefinite programming. We design adecomposition algorithm where smaller-size semidefinite programs can be solvedin parallel, which further reduces the runtime. Lastly, we establish theperformance guarantees of the proposed scheme and demonstrate its effectivenessthrough numerical examples.

A Decision Rule Approach for Two-Stage Data-Driven Distributionally Robust Optimization Problems with Random Recourse