Stochastic programming.

*(English)*Zbl 0812.90122
New York, NY: Wiley. xii, 307 p. (1994).

The purpose of the book is to fill the need for an up-to-date textbook on stochastic programming that would be of interest for a broad circle of readers – students, teachers and researchers not only in mathematics but also in computer science, engineering, business and economy, etc. In the presentation of results, great attention is paid to modeling issues and to interpretation of individual pieces of knowledge and of their relationship. The current progress in numerical techniques for stochastic programming is reflected and algorithms are also described by means of short Pascal-type procedures. Bibliographical notes, exercises and references are provided at the end of each chapter.

The first two chapters can serve as an introduction into optimization. The background material on linear and nonlinear programming contains also a succinct information about algorithms, Chapter 2, “Dynamic Systems”, covers the Bellman principle for solving dynamic programming problems. Stochastic programs are presented from the very beginning as a natural and appropriate way of modelling uncertainty in various mathematical programming problems. Already at the opening of Chapter 1, various approaches how to introduce and to treat properly random parameters in optimization problems are presented and illuminated on a simple example. Subsequently, the general form of the stochastic programming problems is presented and the basic properties of the most frequently used stochastic linear programming models are summarized.

Chapter 2 gives an additional insight into stochastic dynamic models and introduces the concepts of stochastic decision tree, event tree and presents the new scenario aggregation method. Discussion on value of stochastic solution concludes this introductory part.

Chapters 3 and 4 are devoted to a detailed treatment of stochastic linear programs with recourse and problems with probabilistic constraints, respectively. The main emphasis is on solution techniques including approximation and bounds. For instance, Chapter 3 treats in detail the L- shaped method and various bounding techniques and includes sections on the new stochastic decomposition algorithm and on stochastic quasigradient methods. In addition, a method for solving integer two- stage stochastic programs is presented. Chapter 4 introduces solution techniques for problems that assume normal distribution of the coefficients, both in the joint probabilistic constraints and in the individual ones and presents a recent material on bounding general multivariate distribution functions.

The last two chapters deal with special problems, namely, with preprocessing and with network problems for which the results of Chapter 3 assume a simpler form. An application is the stochastic PERT problem.

The first two chapters can serve as an introduction into optimization. The background material on linear and nonlinear programming contains also a succinct information about algorithms, Chapter 2, “Dynamic Systems”, covers the Bellman principle for solving dynamic programming problems. Stochastic programs are presented from the very beginning as a natural and appropriate way of modelling uncertainty in various mathematical programming problems. Already at the opening of Chapter 1, various approaches how to introduce and to treat properly random parameters in optimization problems are presented and illuminated on a simple example. Subsequently, the general form of the stochastic programming problems is presented and the basic properties of the most frequently used stochastic linear programming models are summarized.

Chapter 2 gives an additional insight into stochastic dynamic models and introduces the concepts of stochastic decision tree, event tree and presents the new scenario aggregation method. Discussion on value of stochastic solution concludes this introductory part.

Chapters 3 and 4 are devoted to a detailed treatment of stochastic linear programs with recourse and problems with probabilistic constraints, respectively. The main emphasis is on solution techniques including approximation and bounds. For instance, Chapter 3 treats in detail the L- shaped method and various bounding techniques and includes sections on the new stochastic decomposition algorithm and on stochastic quasigradient methods. In addition, a method for solving integer two- stage stochastic programs is presented. Chapter 4 introduces solution techniques for problems that assume normal distribution of the coefficients, both in the joint probabilistic constraints and in the individual ones and presents a recent material on bounding general multivariate distribution functions.

The last two chapters deal with special problems, namely, with preprocessing and with network problems for which the results of Chapter 3 assume a simpler form. An application is the stochastic PERT problem.

Reviewer: J.Dupačová (Praha)

##### MSC:

90C15 | Stochastic programming |

90-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming |

90-08 | Computational methods for problems pertaining to operations research and mathematical programming |