Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page

Data-Driven Robust Optimization

The last decade witnessed an explosion in the availability of data for operations research applications. Motivated by this growing availability, we propose a novel schema for utilizing data to design uncertainty sets for robust optimization using statistical hypothesis tests. The approach is flexible and widely applicable, and robust optimization problems built from our new sets are computationally tractable, both theoretically and practically. Furthermore, optimal solutions to these problems enjoy a strong, finite-sample probabilistic guarantee. \edit{We describe concrete procedures for choosing an appropriate set for a given application and applying our approach to multiple uncertain constraints. Computational evidence in portfolio management and queuing confirm that our data-driven sets significantly outperform traditional robust optimization techniques whenever data is available.Comment: 38 pages, 15 page appendix, 7 figures. This version updated as of Oct. 201

Robust risk aggregation with neural networks

We consider settings in which the distribution of a multivariate random variable is partly ambiguous. We assume the ambiguity lies on the level of the dependence structure, and that the marginal distributions are known. Furthermore, a current best guess for the distribution, called reference measure, is available. We work with the set of distributions that are both close to the given reference measure in a transportation distance (e.g. the Wasserstein distance), and additionally have the correct marginal structure. The goal is to find upper and lower bounds for integrals of interest with respect to distributions in this set. The described problem appears naturally in the context of risk aggregation. When aggregating different risks, the marginal distributions of these risks are known and the task is to quantify their joint effect on a given system. This is typically done by applying a meaningful risk measure to the sum of the individual risks. For this purpose, the stochastic interdependencies between the risks need to be specified. In practice the models of this dependence structure are however subject to relatively high model ambiguity. The contribution of this paper is twofold: Firstly, we derive a dual representation of the considered problem and prove that strong duality holds. Secondly, we propose a generally applicable and computationally feasible method, which relies on neural networks, in order to numerically solve the derived dual problem. The latter method is tested on a number of toy examples, before it is finally applied to perform robust risk aggregation in a real world instance.Comment: Revised version. Accepted for publication in "Mathematical Finance

Robustness to dependency in portfolio optimization using overlapping marginals

In this paper, we develop a distributionally robust portfolio optimization model where the robustness is across different dependency structures among the random losses. For a Fr´echet class of discrete distributions with overlapping marginals, we show that the distributionally robust portfolio optimization problem is efficiently solvable with linear programming. To guarantee the existence of a joint multivariate distribution consistent with the overlapping marginal information, we make use of a graph theoretic property known as the running intersection property. Building on this property, we develop a tight linear programming formulation to find the optimal portfolio that minimizes the worst-case Conditional Value-at-Risk measure. Lastly, we use a data-driven approach with financial return data to identify the Fr´echet class of distributions satisfying the running intersection property and then optimize the portfolio over this class of distributions. Numerical results in two different datasets show that the distributionally robust portfolio optimization model improves on the sample-based approac

Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity

Cataloged from PDF version of article.We consider the problem of optimal portfolio choice using the Conditional Value-at-Risk (CVaR) and Value-at-Risk (VaR) measures for a market consisting of n risky assets and a riskless asset and where short positions are allowed. When the distribution of returns of risky assets is unknown but the mean return vector and variance/covariance matrix of the risky assets are fixed, we derive the distributionally robust portfolio rules. Then, we address uncertainty (ambiguity) in the mean return vector in addition to distribution ambiguity, and derive the optimal portfolio rules when the uncertainty in the return vector is modeled via an ellipsoidal uncertainty set. In the presence of a riskless asset, the robust CVaR and VaR measures, coupled with a minimum mean return constraint, yield simple, mean-variance efficient optimal portfolio rules. In a market without the riskless asset, we obtain a closed-form portfolio rule that generalizes earlier results, without a minimum mean return restriction

Mean semi-deviation from a target and robust portfolio choice under distribution and mean return ambiguity

Cataloged from PDF version of article.We consider the problem of optimal portfolio choice using the lower partial moments risk measure for a market consisting of n risky assets and a riskless asset. For when the mean return vector and variance/covariance matrix of the risky assets are specified without specifying a return distribution, we derive distributionally robust portfolio rules. We then address potential uncertainty (ambiguity) in the mean return vector as well, in addition to distribution ambiguity, and derive a closed-form portfolio rule for when the uncertainty in the return vector is modelled via an ellipsoidal uncertainty set. Our result also indicates a choice criterion for the radius of ambiguity of the ellipsoid. Using the adjustable robustness paradigm we extend the single-period results to multiple periods, and derive closed-form dynamic portfolio policies which mimic closely the single-period policy. © 2013 Elsevier B.V. All rights reserved