We propose robust optimization models and their tractable approximations that cater for ambiguity-averse decision makers whose underlying risk preferences are consistent with constant absolute risk aversion (CARA). Specifically, we focus on maximizing the worst-case expected exponential utility where the underlying uncertainty is generated from a set of stochastically independent factors with ambiguous marginals. To obtain computationally tractable formulations, we propose a hierarchy of approximations, starting from formulating the objective function as tractable concave functions in affinely perturbed cases, developing approximations in concave piecewise affinely perturbed cases, and proposing new multi-deflected linear decision rules for adaptive optimization models. We also extend the framework to address a multi-period consumption model. The resultant models would take the form of an exponential conic optimization problem (ECOP), which can be practicably solved using current off-the-shelf solvers. We present numerical examples including project management and multi-period inventory management with financing to illustrate how our approach can be applied to obtain high-quality solutions that could outperform current stochastic optimization approaches, especially in situations with high risk aversion levels.
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