Peter Kall

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 methods for solving two-stage stochastic linear programming problems

Approximating a given continuous probability distribution of the data of a linear program by a discrete one yields solution methods for the stochastic linear programming problem with complete fixed recourse. For a procedure along the lines of [8], the reduction of the computational amount of work compared to the usual revised simplex method is figured out. Furthermore, an alternative method is proposed, where by refining particular discrete distributions the optimal value is approximated.ZusammenfassungFür das zweistufige Modell der stochastischen linearen Programmierung mit vollständiger Kompensation werden Verfahren untersucht, die sich aus der Annäherung einer gegebenen stetigen Wahrscheinlichkeitsverteilung der Daten durch endlich diskrete Verteilungen ergeben. Beim Vorgehen nach [8] wird die Reduktion des Rechenaufwandes im Vergleich zur üblichen revidierten Simplexmethode ermittelt. Als Alternative wird ein Verfahren vorgeschlagen, in dem durch sukzessive Verfeinerung speziell gewählter diskreter Verteilungen der Optimalwert monoton angenähert wird.

Solving stochastic programming problems with recourse including error bounds

Under suitable convexity and integrability assumptions, for the stochastic programming problem with recourse statements are proved very easily, which have been shown until now only for stochastic linear programming. In particular, this includes lower bounds for approximations using discrete random vectors. Until now unpublished, even for the linear ease, are error bounds, which are proved here under different assumptions. Computational experiences are reported. Finally, some improvements are suggested which may reduce the computation time.

Approximation to Optimization Problems: An Elementary Review

During the last two decades the concept of epi-convergence was introduced and then was used in various investigations in optimization and related areas. The aim of this review is to show in an elementary way how closely the arguments in the epi-convergence approach are related to those of the classical theory of convergence of functions.

Approximations to stochastic programs with complete fixed recourse

The probability distribution of the data entering a recourse problem is replaced by finite discrete distributions. It is proved that the convergence of the objective functions of the approximating problems to that one of the original problem can be achieved by choosing the discrete distributions in quite a natural way. For bounded feasible sets this implies the convergence of the optimal values. Finally some error bounds are derived.

SLP-IOR: an interactive model management system for stochastic linear programs

In this paper stochastic linear programming (SLP) is considered from the model management point of view. General model management issues specific to SLP are discussed in connection with their implementation in SLP-IOR. The central topic of the paper is SLP-IOR itself which is a model management system for SLP being under development by the authors. The presentation is concentrated on single and two stage models these being the model classes incorporated into the present version of SLP-IOR.

An upper bound for SLP using first and total second moments

In 1987, J. Dulá considered the problem of finding an upper bound for the expectation of a so-called “simplicial” function of a random vector and used for this purpose first and total second moments. Under the same moment conditions we consider some different cases of “recourse” functions and demonstrate how the related moment problems can be solved by solving nonsmooth (unconstrained) optimization problems and thereafter satisfying simple linear constraint systems.

Stochastic programs with recourse: An upper bound and the related moment problem

The probability measurePO on multidimensional intervals in the extension of the Edmundson-Madansky upper bound for stochastically dependent random variables, derived recently in [7], is shown to be the uniquely determined extremal solution of a particular multivariate moment problem. A necessary and sufficient condition for the feasibility of this moment problem is derived, which is shown to coincide for the univariate moment problem with the simplex containing the moment space (see [15]).ZusammenfassungEs wird gezeigt, daß das WahrscheinlichkeitsmaßP0 der kürzlich in [7] gegebenen Verallgemeinerung der Edmundson-Madansky-Schranke für abhängige Zufallsvariable (mit Träger in einem Quader) die eindeutige Lösung eines speziellen multivariaten Momentenproblems ist. Ferner wird eine notwendige und hinreichende Bedingung für die Lösbarkeit dieses Momentenproblems hergeleitet, deren Anwendung auf das klassische univariate Momentenproblem äquivalent ist mit der Forderung, daß der vorgegebene Momentenvektor dem den Momentenraum umfassenden Simplex angehört (vgl. [15]).

Das zweistufige Problem der stochastischen linearen Programmierung

ZusammenfassungEs wird das aus der Literatur ([1], [4], [7], [9]) bekannte zweistufige Problem behandelt, wobei allerdings nicht nur die „rechten Seiten“ bzw. die Koeffizienten der Zielfunktion stochastische Variable sind. ZunÄchst wird das Problem neu formuliert wie in [1], was sich für den Beweis der in [7] und [4] eher umstÄndlich bewiesenen Ungleichungen als nützlich erweist. Schlie\lich wird ein Verfahren der zulÄssigen Richtungen zur Lösung des Problems angegeben und seine Konvergenz bewiesen.

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.