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

Understand

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.