Robust concave utility maximization over chance constraints

Abstract

This paper first studies an expected utility problem with chance constraints and incomplete information on a decision maker’s utility function. The model maximizes the worst-case expected utility of random outcome over a set of concave functions within a novel ambiguity set, while the underlying probability distribution is known with the assumption of a discretization of possible realizations for the uncertainty parameter. To obtain computationally tractable formulations, we employ a discretization approach to derive a max–min chance-constrained approximation of this problem. This approximation is further reformulated as a mixed-integer program. We show that the discrete approximation converges to the true counterpart under mild assumptions. We also present a row generation algorithm for optimizing the max–min program. A computational study for a bin-packing problem and a multi-item newsvendor problem is conducted to demonstrate the benefit of the proposed framework and the computational efficiency of our algorithm. We find that the row generation algorithm can significantly reduce the computational time, and our robust policy could achieve a better out-of-sample performance when compared with the non-robust policy and the one without the chance constraints.