Utility preference robust optimization (PRO) models are recently proposed to deal with decision making problems where the decision maker's true utility function is unknown and the optimal decision is based on the worst case utility function from an ambiguity set of utility functions. In this paper, we consider the case where the ambiguity set is constructed through some moment-type conditions (\cite{HuM15}) and develop a numerical scheme for approximating the ambiguity set so that the resulting maximin optimization problem can be solved for nonconcave utility functions. To justify the approximation scheme, we derive an error bound for the approximated ambiguity set, the optimal value and optimal solutions of the resulting maximin problem. To address the data perturbation/contamination issues in the construction of the ambiguity set, we derive some stability results which quantify the variation of the ambiguity set against perturbation of the elicitation data and its propagation to the optimal value and optimal solutions of the PRO model. Finally, we extend the discussions to the case where the ambiguity set depends on the decision variables and the domain of the utility functions is unbounded. Some preliminary numerical results show that the proposed approximation schemes work very well.

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