Robust mean-covariance solutions for stochastic optimization
Author: Popescu, Ioana INSEAD Area: Decision SciencesIn: Operations Research, vol. 55, no. 1, January/February 2007 Language: EnglishDescription: p. 98-112.Type of document: INSEAD ArticleNote: Please ask us for this itemAbstract: 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 projections property for multivariate distributions with given means and covariances, which reduces our problem to optimizing a univariate mean-variance\textit{robust objective}. This allows us to use known univariate results in the multidimensional settings, and to add new results in this direction. In particular, we characterize a general class of objective functions (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 (1966) 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 multi-product pricing application, which motivates extensions of our results for the case of non-negative and decision dependent returns.| Item type | Current location | Call number | Status | Date due | Barcode | Item holds |
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Europe Campus | Available | BC007854 |
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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 projections property for multivariate distributions with given means and covariances, which reduces our problem to optimizing a univariate mean-variance\textit{robust objective}. This allows us to use known univariate results in the multidimensional settings, and to add new results in this direction. In particular, we characterize a general class of objective functions (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 (1966) 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 multi-product pricing application, which motivates extensions of our results for the case of non-negative and decision dependent returns.
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