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

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

Models and algorithms for dominance-constrained stochastic programs with recourse

In der vorliegenden Dissertationsschrift befassen wir uns mit stochastischen Optimierungsproblemen unter Nebenbedingungen, die mithilfe stochastischer Ordnungen formuliert sind. Hierbei konzentrieren wir uns auf stochastische Dominanz erster Ordnung und die steigende konvexe Ordnung, wobei beide Ordnungen in unserem Fall auf Zufallsgrößen operieren, welche Optimalwerten zweistuger stochastischer Optimierungsprobleme mit Kompensation entsprechen. Wir stellen die theoretische Relevanz der vorliegenden Problemklasse heraus und tragen zur Entwicklung von effizienten Lösungsverfahren bei. Um Letzteres zu erreichen untersuchen und erweitern wir bestehende gemischt-ganzzahlige lineare Repräsentationen dieser Probleme und entwickeln maßgeschneiderte Dekompositionsverfahren. Der Schwerpunkt dieser Arbeit liegt dabei auf der Entwicklung und Implementierung besonders effizienter Lösungsansätze für den Fall mit linearer Kompensation.We consider optimization problems with stochastic order constraints of first and second order posed on random variables coming from two-stage stochastic programs with recourse. We clarify the theoretical relevance of these specific problems, and contribute to improving their computational tractability. For the latter, we review and enhance mixed-integer linear programming (MILP) equivalents. These exist for either mixed-integer or continuous variables in the second stage. Algorithmically, our focus is on developing tailored cutting-plane decomposition methods for these models

Distributionally Robust Optimization: A Review

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization

Exact penalization, level function method, and modified cutting-plane method for stochastic programs with second order stochastic dominance constraints

Level function methods and cutting plane methods have been recently proposed to solve stochastic programs with stochastic second order dominance (SSD) constraints. A level function method requires an exact penalization setup because it can only be applied to the objective function, not the constraints. Slater constraint qualification (SCQ) is often needed for deriving exact penalization. It is well known that SSD usually does not satisfy SCQ and various relaxation schemes have been proposed so that the relaxed problem satisfies the SCQ. In this paper, we show that under some moderate conditions the desired constraint qualification can be guaranteed through some appropriate reformulation of the constraints rather than relaxation. Exact penalization schemes based on L1-norm and L1-norm are subsequently derived through Robinson’s error bound on convex system and Clarke’s exact penalty function theorem. Moreover, we propose a modified cutting plane method which constructs a cutting plane through the maximum of the reformulated constraint functions. In comparison with the existing cutting plane methods, it is numerically more efficient because only a single cutting plane is constructed and added at each iteration. We have carried out a number of numerical experiments and the results show that our methods display better performances particularly in the case when the underlying functions are nonlinear w.r.t. decision variables