Introduction to stochastic programming.

*(English)*Zbl 0892.90142
Springer Series in Operations Research. New York, NY: Springer. xix, 421 p. (1997).

This textbook on stochastic programming is devoted to the description of the main methods so far developed in this area. It is highly connected with modern applications, in particular with those of economic nature. The authors try to describe several things: objects, models, mathematical brackground, theoretical study (exact solution), numerical study (approximate solution). Most of the prerequisites and basic notions and results are given in the book. The reader can get an opinion not only on how stochastic analysis works, but also find his own way to treat concrete problems using stochastic programming methods.

The book is divided into five parts related to different sides of this theory. Most of the results are illustrated by modelling examples. A series of exercises is also helpful to understand the main ideas of the book. The large bibliography can be useful for students, teachers, and for those trying to use this material in practical applications.

The first part, “Models”, is concerned with applications examples and models which lead to stochastic programming problems. Special attention is paid to the uncertainty of data and to the randomization of the standard deterministic questions.

In the second part, “Basic Properties”, the theory of stochastic programming is developed (more precisely, it deals with problems’ formulation, their classification, properties of model equations, as well as of solutions). Among the main ideas described are those about the expected value of perfect information and the expected value of the stochastic solution.

Part 3, “Solution Methods”, gives a description of the methods of stochastic programming. They are gathered in four chapters devoted to different types of problems: two-stage linear recourse problems, two-stage nonlinear recourse problems, multistage problems, stochastic integer problems.

In part 4, “Approximating and Sampling Methods”, some approximative approaches and numerical methods are studied. Among them are evaluation and approximate expectation, Monte Carlo methods and multistage approximation.

The short part 5, “A Case Study”, presents an example of stochastic programming consisting of models with flexible capacity.

The book is divided into five parts related to different sides of this theory. Most of the results are illustrated by modelling examples. A series of exercises is also helpful to understand the main ideas of the book. The large bibliography can be useful for students, teachers, and for those trying to use this material in practical applications.

The first part, “Models”, is concerned with applications examples and models which lead to stochastic programming problems. Special attention is paid to the uncertainty of data and to the randomization of the standard deterministic questions.

In the second part, “Basic Properties”, the theory of stochastic programming is developed (more precisely, it deals with problems’ formulation, their classification, properties of model equations, as well as of solutions). Among the main ideas described are those about the expected value of perfect information and the expected value of the stochastic solution.

Part 3, “Solution Methods”, gives a description of the methods of stochastic programming. They are gathered in four chapters devoted to different types of problems: two-stage linear recourse problems, two-stage nonlinear recourse problems, multistage problems, stochastic integer problems.

In part 4, “Approximating and Sampling Methods”, some approximative approaches and numerical methods are studied. Among them are evaluation and approximate expectation, Monte Carlo methods and multistage approximation.

The short part 5, “A Case Study”, presents an example of stochastic programming consisting of models with flexible capacity.

Reviewer: S.V.Rogozin (Minsk)