DISTRIBUTIONALLY ROBUST OPTIMIZATION: A COMPREHENSIVE SURVEY OF THEORY, METHODS, AND APPLICATIONS
Abstract
Decision-making under uncertainty is one of the most stubborn challenges in engineering, economics, finance, and operations research. Traditional stochastic programming usually starts from the bold idea that we already know the exact probability distribution, which, if we are honest, almost never happens in the real world. In practice, data are messy, incomplete, sometimes even contradictory, yet the models pretend otherwise. Distributionally robust optimization, or DRO as it’s now called, tries to hedge against this kind of distributional confusion by optimizing the worst-case expected performance over a set of believable, or at least plausible, probability models this set is what people call the ambiguity set. This survey tries to give a somewhat systematic, though not exhaustive, look at the theory, methods, and uses of DRO over roughly the past fifteen years, which is a long time in optimization years. We arrange the discussion around three big questions that keep coming back: how to build ambiguity sets when data are scarce or partial, what makes a DRO problem actually solvable on a computer, and how it performs in practice compared to robust optimization or the old stochastic programming. Our reading of the literature suggests that modern DRO, especially the versions based on the Wasserstein distance, manages to balance statistical reliability with computational effort in a way that feels, well, reasonable. It doesn’t overreact but it doesn’t ignore risk either. We also point to some directions that look promising but still uncertain multi-stage formulations that evolve over time, tighter links with machine learning models that keep changing, and applications that matter socially like climate resilience or personalized medicine. The survey is meant both as a map for newcomers who feel lost in the forest of DRO papers and as a kind of consolidated reference for researchers already working in the area who want a single place to see what’s been happening, even if not everything fits neatly together.
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