Robust Optimization of Rank-Dependent Models with Uncertain Probabilities

This paper studies distributionally robust optimization for a rich class of risk measures with ambiguity sets defined by $\phi$-divergences. The risk measures are allowed to be non-linear in probabilities, are represented by Choquet integrals possibly induced by a probability weighting function, and encompass many well-known examples. Optimization for this class of risk measures is challenging due to their rank-dependent nature. We show that for various shapes of probability weighting functions, including concave, convex and inverse $S$-shaped, the robust optimization problem can be reformulated into a rank-independent problem. In the case of a concave probability weighting function, the problem can be reformulated further into a convex optimization problem that admits explicit conic representability for a collection of canonical examples. While the number of constraints in general scales exponentially with the dimension of the state space, we circumvent this dimensionality curse and develop two types of algorithms. They yield tight upper and lower bounds on the exact optimal value and are formally shown to converge asymptotically. This is illustrated numerically in a robust newsvendor problem and a robust portfolio choice problem.