Decision tools in risk management figure amongst the oldest applications of numerical optimization. Optimization models that arise in this context typically rely on model parameters. In practical applications these parameters have to be estimated and are therefore not known with certainty. To mitigate the effects of instability of optimal investment decisions as a function of the model parameters, robust optimization aims at finding solutions that behave well for all points in an uncertainty set for the model parameters. The existing literature thereby largely treats uncertainty in parameters that model risk independently of parameters that model expected returns. We argue that - in the typical situation where return data cannot be independently sampled but is available through historical data - functional dependencies between the risk and expected return terms of the model parameters arise naturally, leading to structured uncertainty sets that are smaller and less pessimistic than the standard models considered in the literature. We show that several new portfolio optimization models based on structured uncertainty sets that reside in quadratic manifolds have equivalent reformulations as convex quadratic programming problems. Our numerical results based on real market data suggest that the practical performance of our new models compares favorably with existing methods.
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