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

TL;DR

This paper introduces a scalable, data-driven distributionally robust approach for two-stage stochastic optimization with random recourse, utilizing decision rules and copositive reformulations to improve tractability and provide performance guarantees.

Contribution

It proposes a novel approximation scheme using piecewise decision rules and a copositive reformulation for two-stage robust optimization with random recourse, enhancing computational efficiency.

Findings

The method effectively handles distributional uncertainty with two layers of robustness.

The copositive reformulation allows for tractable semidefinite programming approximations.

Numerical examples demonstrate improved computational performance and solution quality.

Abstract

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.