Optimization with multivariate conditional value-at-risk constraints

For many decision making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice. As an alternative, we focus on the widely-applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces we develop a cut generation algorithm, where each cut is obtained by solving a mixed integer problem. We show that a multivariate CVaR constraint reduces to finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible, and computationally tractable way of modeling preferences in stochastic multi-criteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the proposed solution methods

Optimization with multivariate conditional value-at-risk constraints

For many decision making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice. As an alternative, we focus on the widely-applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces we develop a cut generation algorithm, where each cut is obtained by solving a mixed integer problem. We show that a multivariate CVaR constraint reduces to finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible, and computationally tractable way of modeling preferences in stochastic multi-criteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the proposed solution methods

Reduced Complexity Filtering with Stochastic Dominance Bounds: A Convex Optimization Approach

This paper uses stochastic dominance principles to construct upper and lower sample path bounds for Hidden Markov Model (HMM) filters. Given a HMM, by using convex optimization methods for nuclear norm minimization with copositive constraints, we construct low rank stochastic marices so that the optimal filters using these matrices provably lower and upper bound (with respect to a partially ordered set) the true filtered distribution at each time instant. Since these matrices are low rank (say R), the computational cost of evaluating the filtering bounds is O(XR) instead of O(X2). A Monte-Carlo importance sampling filter is presented that exploits these upper and lower bounds to estimate the optimal posterior. Finally, using the Dobrushin coefficient, explicit bounds are given on the variational norm between the true posterior and the upper and lower bounds

Stochastic programming models and methods for portfolio optimization and risk management

This project is focused on stochastic models and methods and their application in portfolio optimization and risk management. In particular it involves development and analysis of novel numerical methods for solving these types of problem. First, we study new numerical methods for a general second order stochastic dominance model where the underlying functions are not necessarily linear.Specifically, we penalize the second order stochastic dominance constraints to the objective under Slater’s constraint qualification and then apply the well known stochastic approximation method and the level function methods to solve the penalized problem and present the corresponding convergence analysis. All methods are applied to some portfolio optimization problems, where the underlying functions are not necessarily linear all results suggests that the portfolio strategy generated by the second order stochastic dominance model outperform the strategy generated by the Markowitz model in a sense of having higher return and lower risk. Furthermore a nonlinear supply chain problem is considered, where the performance of the level function method is compared to the cutting plane method. The results suggests that the level function method is more efficient in a sense of having lower CPU time as well as being less sensitive to the problem size. This is followed by study of multivariate stochastic dominance constraints. We propose a penalization scheme for the multivariate stochastic dominance constraint and present the analysis regarding the Slater constraint qualification. The penalized problem is solved by the level function methods and a modified cutting plane method and compared to the cutting surface method proposed in [70] and the linearized method proposed in [4]. The convergence analysis regarding the proposed algorithms are presented. The proposed numerical schemes are applied to a generic budget allocation problem where it is shown that the proposed methods outperform the linearized method when the problem size is big. Moreover, a portfolio optimization problem is considered where it is shown that the a portfolio strategy generated by the multivariate second order stochastic dominance model outperform the portfolio strategy generated by the Markowitz model in sense of having higher return and lower risk. Also the performance of the algorithms is investigated with respect to the computation time and the problem size. It is shown that the level function method and the cutting plane method outperform the cutting surface method in a sense of both having lower CPU time as well as being less sensitive to the problem size. Finally, reward-risk analysis is studied as an alternative to stochastic dominance. Specifically, we study robust reward-risk ratio optimization. We propose two robust formulations, one based on mixture distribution, and the other based on the first order moment approach. We propose a sample average approximation formulation as well as a penalty scheme for the two robust formulations respectively and solve the latter with the level function method. The convergence analysis are presented and the proposed models are applied to Sortino ratio and some numerical test results are presented. The numerical results suggests that the robust formulation based on the first order moment results in the most conservative portfolio strategy compared to the mixture distribution model and the nominal model

Portfolio choice and estimation risk : a comparison of Bayesian approaches to resampled efficiency

Estimation risk is known to have a huge impact on mean/variance (MV) optimized portfolios, which is one of the primary reasons to make standard Markowitz optimization unfeasible in practice. Several approaches to incorporate estimation risk into portfolio selection are suggested in the earlier literature. These papers regularly discuss heuristic approaches (e.g., placing restrictions on portfolio weights) and Bayesian estimators. Among the Bayesian class of estimators, we will focus in this paper on the Bayes/Stein estimator developed by Jorion (1985, 1986), which is probably the most popular estimator. We will show that optimal portfolios based on the Bayes/Stein estimator correspond to portfolios on the original mean-variance efficient frontier with a higher risk aversion. We quantify this increase in risk aversion. Furthermore, we review a relatively new approach introduced by Michaud (1998), resampling efficiency. Michaud argues that the limitations of MV efficiency in practice generally derive from a lack of statistical understanding of MV optimization. He advocates a statistical view of MV optimization that leads to new procedures that can reduce estimation risk. Resampling efficiency has been contrasted to standard Markowitz portfolios until now, but not to other approaches which explicitly incorporate estimation risk. This paper attempts to fill this gap. Optimal portfolios based on the Bayes/Stein estimator and resampling efficiency are compared in an empirical out-of-sample study in terms of their Sharpe ratio and in terms of stochastic dominance

Pareto efficiency for the concave order and multivariate comonotonicity

In this paper, we focus on efficient risk-sharing rules for the concave dominance order. For a univariate risk, it follows from a comonotone dominance principle, due to Landsberger and Meilijson [25], that efficiency is characterized by a comonotonicity condition. The goal of this paper is to generalize the comonotone dominance principle as well as the equivalence between efficiency and comonotonicity to the multi-dimensional case. The multivariate setting is more involved (in particular because there is no immediate extension of the notion of comonotonicity) and we address it using techniques from convex duality and optimal transportation