Distributionally robust optimization (DRO) has arose as an important paradigm to address the issue of distributional ambiguity in decision optimization. In its standard form, DRO seeks an optimal solution against the worst-possible expected value evaluated based on a set of candidate distributions. In the case where a decision maker is not risk neutral, the most common scheme applied in DRO to capture one's risk attitude is employing an expected utility functional. In this paper, we propose to address a decision maker's risk attitude in DRO by following an alternative scheme known as ``dual expected utility''. In this scheme, a distortion function is applied to convert physical probabilities to subjective probabilities so that the resulting expectation, also known as distorted expectation, captures the decision maker's risk attitude. Unlike an expected utility functional which is linear in probability, in the dual scheme a distorted expectation is generally non-linear in probability. We distinguish DRO based on distorted expectations by terming it ``Distributionally Robust Risk Optimization'' (DRRO), and show that DRRO can be equally, if not more, tractable to solve than DRO based on utility functionals. Our tractability results hold for any distortion function, and hence our scheme provides more flexibility to capture more realistic forms of risk attitudes. These include, as an important example, the inverse S-shaped distortion functionals that play a prominent role in Cumulative Prospect Theory (CPT), and several other non-convex risk measures developed more recently. Central to our development is the characterization of worst-case distributions based on the notion of convex envelope, which enables us to discover ``hidden convexity" in DRRO. We demonstrate through a numerical example that a production manager who overly weights ``very good" and ``very bad" outcomes may act as if (s)he is risk-averse when taking into account distributional ambiguity. Worst-case distributions are presented that can provide further explanation of such risk-averse behaviour.
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