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

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.Comment: Previous title: Mathematical Foundations of Robust and Distributionally Robust Optimizatio

Data-driven Approximation of Distributionally Robust Chance Constraints using Bayesian Credible Intervals

The non-convexity and intractability of distributionally robust chance constraints make them challenging to cope with. From a data-driven perspective, we propose formulating it as a robust optimization problem to ensure that the distributionally robust chance constraint is satisfied with high probability. To incorporate available data and prior distribution knowledge, we construct ambiguity sets for the distributionally robust chance constraint using Bayesian credible intervals. We establish the congruent relationship between the ambiguity set in Bayesian distributionally robust chance constraints and the uncertainty set in a specific robust optimization. In contrast to most existent uncertainty set construction methods which are only applicable for particular settings, our approach provides a unified framework for constructing uncertainty sets under different marginal distribution assumptions, thus making it more flexible and widely applicable. Additionally, under the concavity assumption, our method provides strong finite sample probability guarantees for optimal solutions. The practicality and effectiveness of our approach are illustrated with numerical experiments on portfolio management and queuing system problems. Overall, our approach offers a promising solution to distributionally robust chance constrained problems and has potential applications in other fields

Dynamic asset allocation with uncertain jump risks : a pathwise optimization approach

This paper studies the dynamic portfolio choice problem with ambiguous jump risks in a multi-dimensional jump-diffusion framework. We formulate a continuous-time model of incomplete market with uncertain jumps. We develop an efficient pathwise optimization procedure based on the martingale methods and minimax results to obtain closed-form solutions for the indirect utility function and the probability of the worst scenario. We then introduce an orthogonal decomposition method for the multi-dimensional problem to derive the optimal portfolio strategy explicitly under ambiguity aversion to jump risks. Finally, we calibrate our model to real market data drawn from ten international indices and illustrate our results by numerical examples. The certainty equivalent losses affirm the importance of jump uncertainty in optimal portfolio choice