Csaba I. Fábián

Handling CVaR objectives and constraints in two-stage stochastic models

Based on the polyhedral representation of Kunzi-Bay and Mayer [Kunzi-Bay, A., Mayer, J., 2006. Computational aspects of minimizing conditional value-at-risk. Computational Management Science 3, 3-27] , we propose decomposition frameworks for handling CVaR objectives and constraints in two-stage stochastic models. For the solution of the decomposed problems we propose special Level-type methods.

Algorithms for handling CVaR constraints in dynamic stochastic programming

We propose dual decomposition and solution schemes for multistage CVaR-constrained problems. These schemes meet the need for handling multiple CVaR-constraints for different time frames and at different confidence levels. Hence they allow shaping distributions according to the decision maker’s preferences. With minor modifications, the proposed schemes can be used to decompose further types of risk constraints in dynamic portfolio management problems. We consider integrated chance constraints, second-order stochastic dominance constraints, and constraints involving a special value-of-information risk measure. We also suggest application to further financial problems. We propose a dynamic riskconstrained optimization model for option pricing. Moreover we propose special mid-term constraints for use in asset-liability management.

An enhanced model for portfolio choice with SSD criteria: a constructive approach

We formulate a portfolio planning model that is based on second-order stochastic dominance as the choice criterion. This model is an enhanced version of the multi-objective model proposed by Roman et al. [Math. Progr. Ser. B, 2006, 108, 541–569]; the model compares the scaled values of the different objectives, representing tails at different confidence levels of the resulting distribution. The proposed model can be formulated as a risk minimization model where the objective function is a convex risk measure; we characterize this risk measure and the resulting optimization problem. Moreover, our formulation offers a natural generalization of the SSD-constrained model of Dentcheva and Ruszczyński [J. Bank. Finance, 2006, 30, 433–451]. A cutting plane-based solution method for the proposed model is outlined. We present a computational study showing: (a) the effectiveness of the solution methods and (b) the improved modeling capabilities: the resulting portfolios have superior return distributions.

A computational study of a solver system for processing two-stage stochastic LPs with enhanced Benders decomposition

We report a computational study of two-stage SP models on a large set of benchmark problems and consider the following methods: (i) Solution of the deterministic equivalent problem by the simplex method and an interior point method, (ii) Benders decomposition (L-shaped method with aggregated cuts), (iii) Regularised decomposition of Ruszczyński (Math Program 35:309–333, 1986), (iv) Benders decomposition with regularization of the expected recourse by the level method (Lemaréchal et al. in Math Program 69:111–147, 1995), (v) Trust region (regularisation) method of Linderoth and Wright (Comput Optim Appl 24:207–250, 2003). In this study the three regularisation methods have been introduced within the computational structure of Benders decomposition. Thus second-stage infeasibility is controlled in the traditional manner, by imposing feasibility cuts. This approach allows extensions of the regularisation to feasibility issues, as in Fábián and Szőke (Comput Manag Sci 4:313–353, 2007). We report computational results for a wide range of benchmark problems from the POSTS and SLPTESTSET collections and a collection of difficult test problems compiled by us. Finally the scale-up properties and the performance profiles of the methods are presented.

Algorithms for handling CVaR-constraints in dynamic stochastic programming models with applications to finance

We propose dual decomposition and solution schemes for multistage CVaR-constrained problems. These schemes meet the need for handling multiple CVaR-constraints for different time frames and at different confidence levels. Hence they allow shaping distributions according to the decision makers preference.With minor modifications, the proposed schemes can be used to decompose further types of risk constraints in dynamic portfolio management problems. We consider integrated chance constraints, second-order stochastic dominance constraints, and constraints involving a special value-of-information risk measure. We also suggest application to further financial problems. We propose a dynamic risk-constrained optimization model for option pricing. Moreover we propose special mid-term constraints for use in asset-liability management.

A computational study of a solver system for processing two-stage stochastic linear programming problems

Formulation of stochastic optimisation problems and computational algorithms for their solution continue to make steady progress as can be seen from an analysis of many developments in this field. The edited volume by Wallace and Ziemba (2005) outlines both the modelling systems for stochastic programming (SP) and many applications in diverse domains. More recently, Fabozzi et al. (2007) have considered the application of SP models to challenging financial engineering problems. The tightly knit yet highly focused Stochastic Programming Community, their active website http://stoprog.org, and their triennial international SP conference points to the progressive acceptance of SP as a valuable decision tool. The Committee on Stochastic Programming (COSP) exists as a standing committee of the Mathematical Optimization Society, and also serves as a liaison to related professional societies to promote stochastic programming. At the same time many of the major software vendors, namely, XPRESS, AIMMS, MAXIMAL, and GAMS have started offering SP extensions to their

Risk‐averse optimization in two‐stage stochastic models: computational aspects and a study

We extend the on-demand accuracy approach of Oliveira and Sagastizabal to constrained convex optimization. The resulting method is applied to risk-averse two-stage stochastic programming problems. We present a survey of risk-averse models. The appropriate oracle is formulated for the case of a conditional value-at-risk constraint. We discuss computational aspects and compare different approaches in a study.

Applying oracles of on-demand accuracy in two-stage stochastic programming – A computational study

Traditionally, two variants of the L-shaped method based on Benders’ decomposition principle are used to solve two-stage stochastic programming problems: the aggregate and the disaggregate version. In this study we report our experiments with a special convex programming method applied to the aggregate master problem. The convex programming method is of the type that uses an oracle with on-demand accuracy. We use a special form which, when applied to two-stage stochastic programming problems, is shown to integrate the advantages of the traditional variants while avoiding their disadvantages. On a set of 105 test problems, we compare and analyze parallel implementations of regularized and unregularized versions of the algorithms. The results indicate that solution times are significantly shortened by applying the concept of on-demand accuracy.

Portfolio Choice Models Based on Second-Order Stochastic Dominance Measures: An Overview and a Computational Study

In this chapter we present an overview of second-order stochastic dominance-based models with a focus on those using dominance measures. In terms of portfolio policy, the aim is to find a portfolio whose return distribution dominates the index distribution to the largest possible extent. We compare two approaches, the unscaled model of Roman et al. (Mathematical Programming Series B 108: 541–569, 2006) and the scaled model of Fabian et al. (Quantitative Finance 2010). We constructed optimal portfolios using representations of the future asset returns given by historical data on the one hand, and scenarios generated by geometric Brownian motion on the other hand. In the latter case, the parameters of the GBM were obtained from the historical data. Our test data consisted of stock returns from the FTSE 100 basket, together with the index returns. Part of the data were reserved for out-of-sample tests. We examined the return distributions belonging to the respective optimal portfolios of the unscaled and the scaled problems. The unscaled model focuses on the worst cases and hence enhances safety. We found that the performance of the unscaled model is improved by using scenario generators. On the other hand, the scaled model replicates the shape of the index distribution. Scenario generation had little effect on the scaled model. We also compared the shapes of the histograms belonging to corresponding pairs of in-sample and out-of-sample tests and observed a remarkable robustness in both models. We think these features make these dominance measures good alternatives for classic risk measures in certain applications, including certain multistage ones. We mention two candidate applications.

On a dual method for a specially structured linear programming problem with application to stochastic programming

This article revises and improves on a Dual Type Method (DTM), developed by Prékopa. (Prékopa, A. (1990). Dual method for the solution of a one-stage stochastic programming problem with random RHS obeying a discrete probability distribution, ZOR-Methods and Models of Operations Research , 34 , 441-461), in two ways. The first one allows us, in each iteration, to perform the largest step toward the optimum. The second one consists of exploiting the structure of the working basis , which has to be inverted in each iteration, and updating its inverse in product form, as it is usual in case of the standard dual method. The improved method has been implemented. A report on its performance on the solution of some stochastic programming problems is also presented.