Distributionally robust optimal uncertainty quantification under Phi-divergence ambiguity

We investigate distributionally robust optimal uncertainty quantification (DROUQ) problems under Phi-divergence ambiguity. The DROUQ model is a semi-infinite fractional distributionally robust optimization problem with an infinite number of adaptive chance constraints. The objective is to maximize the worst-case probability that a response function, expressed as the ratio of two stochastic functions, exceeds a given performance threshold. We propose a generic reformulation, which transforms the DROUQ problem – under any Phi-divergence – into a finite-dimensional nonconvex chance-constrained problem. Building on this result and focusing on the Variation and the Hellinger distance Phi-divergences, we further demonstrate that the DROUQ model can be reformulated as a biconvex continuous second-order cone problem for several types of nonlinear fractional response functions. We design a new monotonic alternative convex search algorithm that converges finitely and solves exactly the biconvex reformulations. We evaluate our approach on a financial problem in which the response function is the Sharpe ratio of a portfolio and the objective is to determine the investment strategy that maximizes the worst-case probability under which the Sharpe ratio reaches the required level. The tests attest that the algorithm is computationally efficient and that the solution time is invariant with the size of the nominal distribution. The out-of-sample tests show that the DROUQ models outperform the ambiguity-free one.