The insights from this work can be extended to stochastic programming (SP) problems with uncertain constraints or nonconvex objectives by leveraging the established sample complexity bounds that are free from metric entropy. The findings indicate that the Sample Average Approximation (SAA) method can maintain its efficacy even in scenarios where traditional Lipschitz conditions do not hold. This flexibility allows for the adaptation of SAA to handle uncertain constraints, which often arise in real-world applications where constraints may be subject to variability or incomplete information.
For nonconvex objectives, the results suggest that SAA can still provide reliable sample complexity bounds, particularly in cases where the underlying randomness is heavy-tailed or exhibits non-Lipschitzian behavior. By focusing on the properties of the objective function and the associated gradients, practitioners can formulate SAA approaches that accommodate the complexities of nonconvex landscapes. This could involve developing regularization techniques similar to those discussed in the paper, which can help stabilize the optimization process and ensure convergence to satisfactory solutions.
Moreover, the theoretical framework established in this work can serve as a foundation for future research aimed at exploring the interplay between sample complexity and the structural properties of SP problems, including those with uncertain constraints and nonconvex objectives. By systematically analyzing these relationships, researchers can derive new algorithms that are both efficient and robust in the face of uncertainty.