János Mayer

Stochastic linear programming

Peter Kall and Janos Mayer are distinguished scholars and professors of Operations Research and their research interest is particularly devoted to the area of stochastic optimization. Stochastic Linear Programming: Models, Theory, and Computation is a definitive presentation and discussion of the theoretical properties of the models, the conceptual algorithmic approaches, and the computational issues relating to the implementation of these methods to solve problems that are stochastic in nature. The application area of stochastic programming includes portfolio analysis, financial optimization, energy problems, random yields in manufacturing, risk analysis, etc. In this book, models in financial optimization and risk analysis are discussed as examples, including solution methods and their implementation. Stochastic programming is a fast developing area of optimization and mathematical programming. Numerous papers and conference volumes, and several monographs have been published in the area; however, the Kall and Mayer book will be particularly useful in presenting solution methods including their solid theoretical basis and their computational issues, based in many cases on implementations by the authors. The book is also suitable for advanced courses in stochastic optimization.

Computational aspects of minimizing conditional value-at-risk

Abstract.We consider optimization problems for minimizing conditional value-at-risk (CVaR) from a computational point of view, with an emphasis on financial applications. As a general solution approach, we suggest to reformulate these CVaR optimization problems as two-stage recourse problems of stochastic programming. Specializing the L-shaped method leads to a new algorithm for minimizing conditional value-at-risk. We implemented the algorithm as the solver CVaRMin. For illustrating the performance of this algorithm, we present some comparative computational results with two kinds of test problems. Firstly, we consider portfolio optimization problems with 5 random variables. Such problems involving conditional value at risk play an important role in financial risk management. Therefore, besides testing the performance of the proposed algorithm, we also present computational results of interest in finance. Secondly, with the explicit aim of testing algorithm performance, we also present comparative computational results with randomly generated test problems involving 50 random variables. In all our tests, the experimental solver, based on the new approach, outperformed by at least one order of magnitude all general-purpose solvers, with an accuracy of solution being in the same range as that with the LP solvers.

Existence of Sunspot Equilibria and Uniqueness of Spot Market Equilibria: The Case of Intrinsically Complete Markets

We consider economies with additively separable utility functions and give conditions for the two-agents case under which the existence of sunspot equilibria is equivalent to the occurrence of the transfer paradox. This equivalence enables us to show that sunspots cannot matter if the initial economy has a unique spot market equilibrium and there are only two commodities or if the economy has a unique equilibrium for all distributions of endowments induced by asset trade. For more than two agents the equivalence breaks and we give an example for sunspot equilibria even though the economy has a unique equilibrium for all distributions of endowments induced by asset trade.

Linearly constrained estimation by mathematical programming

Abstract Some mathematical programming models of the mixing problem are discussed in this paper. Five models, based on different discrepancies, are considered and their fundamental properties are examined. Using variational and Smirnov distances, linear programming models are obtained. Pearson divergence leads to quadratic programming, Hellinger divergence leads to l p -programming and the Kullback-Leibler information divergence gives a special geometric programming model. Finally some computational experiences are presented.

6. Building and Solving Stochastic Linear Programming Models with SLP-IOR

The goal of this chapter is to describe the capabilities and the usage of SLP–IOR, our interactive model management system for stochastic linear programming (SLP). The main features of SLP–IOR are the following: the system is intended to support the entire life cycle of a model, including model formulation, analysis of the model instance, solving it, and analyzing the solution. A main design characteristic is keeping connection to an algebraic modeling system; we have chosen GAMS (Brooke, Kendrick, and Meeraus 1992, Brooke et al. 1998). This approach has the following advantages: on the one hand, the powerful general–purpose solvers connected to GAMS are available for solving deterministic equivalents of SLP problems. On the other hand, deterministic LP’s formulated in the algebraic modeling language of GAMS can be imported into SLP–IOR for the purpose of developing stochastic variants of these. However, the usage of GAMS is optional; with the exception of the above–mentioned GAMS–related features, SLP–IOR can be fully utilized without having access to GAMS.

On the Numerical Solution of Jointly Chance Constrained Problems

This paper considers jointly chance constrained problems from the numerical point of view. The main numerical difficulties as well as techniques for overcoming these difficulties are discussed. The efficiency of the approach is illustrated by presenting computational results for large-scale jointly chance constrained test problems.

On Testing SLP Codes with SLP-IOR

This paper gives a summary of selected testing features of the model management system SLP-IOR. The pseudo random test problem generator GENSLP is described and numerical examples with randomly generated test problems are presented.

Cumulative Prospect Theory and Mean Variance Analysis: A Rigorous Comparison

We compare asset allocations that are derived for cumulative prospect theory (CPT) based on two different methods: maximizing CPT along the mean {variance efficient frontier and maximizing CPT without this restriction. We find that with normally distributed returns, the difference between these two approaches is negligible. However, if standard asset allocation data for pension funds are considered, the difference is considerable. Moreover, for certain types of derivatives, such as call options, the restriction of asset allocations to the mean-variance efficient frontier produces sizable losses in various respects, including decreases in expected returns and expected utility.

A note on reward-risk portfolio selection and two-fund separation

This paper presents a general reward-risk portfolio selection model and derives sufficient conditions for two-fund separation. In particular we show that many reward-risk models presented in the literature satisfy these conditions.

Some insights into the solution algorithms for SLP problems

We consider classes of stochastic linear programming problems which can be efficiently solved by deterministic algorithms. For two–stage recourse problems we identify two such classes. The first one consists of problems where the number of stochastically independent random variables is relatively low; the second class is the class of simple recourse problems. The proposed deterministic algorithm is successive discrete approximation. We also illustrate the impact of required accuracy on the efficiency of this algorithm. For jointly chance constrained problems with a random right–hand–side and multivariate normal distribution we demonstrate the increase in efficiency when lower accuracy is required, for a central cutting plane method. We support our argumentation and findings with computational results.