# A Unified Theory of Robust and Distributionally Robust Optimization via the Primal-Worst-Equals-Dual-Best Principle

arXiv: https://arxiv.org/abs/2105.00760 · HTML: https://mlsanity.com/paper/2105.00760/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 May 2021.

## Built on by (most cited first, 13 of 13)

- [Outlier-Robust Wasserstein DRO](https://mlsanity.com/paper/2311.05573/) (2023)
- [A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2205.00362/) (2022)
- [On the convex formulations of robust Markov decision processes](https://mlsanity.com/paper/2209.10187/) (2022)
- [Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization](https://mlsanity.com/paper/2303.03900/) (2023)
- [Causality-Inspired Robustness for Nonlinear Models via Representation Learning](https://mlsanity.com/paper/2505.12868/) (2025)
- [Distributionally Robust Multi-Agent Reinforcement Learning for Dynamic Chute Mapping](https://mlsanity.com/paper/2503.09755/) (2025)
- [Wasserstein Distributionally Robust Regret Optimization](https://mlsanity.com/paper/2504.10796/) (2025)
- [Iterative Sampling Methods for Sinkhorn Distributionally Robust Optimization](https://mlsanity.com/paper/2512.12550/) (2025)
- [Transition Uncertainties in Constrained Markov Decision Models: A Robust Optimization Approach](https://mlsanity.com/paper/2503.12238/) (2025)
- [Wasserstein Distributionally Robust Optimization Through the Lens of Structural Causal Models and Individual Fairness](https://mlsanity.com/paper/2509.26275/) (2025)
- [Concave Certificates: Geometric Framework for Distributionally Robust Risk and Complexity Analysis](https://mlsanity.com/paper/2601.01311/) (2026)
- [Smart predict-then-robustly-optimize](https://mlsanity.com/paper/2607.21773/) (2026)
- [The Value of Human Expertise](https://mlsanity.com/paper/2608.26051/) (2026)

## Abstract (arXiv)

Robust and distributionally robust optimization are modeling paradigms for decision-making under uncertainty where the uncertain parameters are only known to reside in an uncertainty set or are governed by any probability distribution from within an ambiguity set, respectively, and a decision is sought that minimizes a cost function under the most adverse outcome of the uncertainty. In this paper, we develop a rigorous and general theory of robust and distributionally robust nonlinear optimization using the language of convex analysis. Our framework is based on a generalized `primal-worst-equals-dual-best' principle that establishes strong duality between a semi-infinite primal worst and a non-convex dual best formulation, both of which admit finite convex reformulations. This principle offers an alternative formulation for robust optimization problems that obviates the need to mobilize the machinery of abstract semi-infinite duality theory to prove strong duality in distributionally robust optimization. We illustrate the modeling power of our approach through convex reformulations for distributionally robust optimization problems whose ambiguity sets are defined through general optimal transport distances, which generalize earlier results for Wasserstein ambiguity sets.