Robust mean-covariance solutions for stochastic optimization

Robust mean-covariance solutions for stochastic optimization

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Article ID: iaor20081494
Country: United States
Volume: 55
Issue: 1
Start Page Number: 98
End Page Number: 112
Publication Date: Jan 2007
Journal: Operations Research
Authors:
Abstract:

We provide a method for deriving robust solutions to certain stochastic optimization problems, based on mean-covariance information about the distributions underlying the uncertain vector of returns. We prove that for a general class of objective functions, the robust solutions amount to solving a certain deterministic parametric quadratic program. We first prove a general projection property for multivariate distributions with given means and covariances, which reduces our problem to optimizing a univariate mean–variance robust objective. This allows us to use known univariate results in the multidimensional setting, and to add new results in this direction. In particular, we characterize a general class of objective functions (the so-called one- or two-point support functions), for which the robust objective is reduced to a deterministic optimization problem in one variable. Finally, we adapt a result from Geoffrion to reduce the main problem to a parametric quadratic program. In particular, our results are true for increasing concave utilities with convex or concave–convex derivatives. Closed-form solutions are obtained for special discontinuous criteria, motivated by bonus- and commission-based incentive schemes for portfolio management. We also investigate a multiproduct pricing application, which motivates extensions of our results for the case of nonnegative and decision-dependent returns.

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