Stochastic programming.

*(English)*Zbl 0834.90098
Mathematics and its Applications (Dordrecht). 324. Dordrecht: Kluwer Academic Publishers. xviii, 599 p. (1995).

This monograph is an important contribution to the rapidly developing field of stochastic programming. It is the second book on stochastic programming to appear recently. (The other one is ‘Stochastic programming’ by P. Kall and S. W. Wallace (1994; Zbl 0812.90122).) The book under review differs mainly in its focus on more advanced readers and by its broader scope determined by the author’s definition of the subject: In addition to the customary definition of stochastic programming as the science that “handles mathematical programming problems, where some of the parameters are random variables”, the author suggests to define it also as the science “which offers solutions for problems formulated in connection with stochastic systems, where the resulting numerical problem to be solved is a mathematical programming problem of a nontrivial size”. Reflections to statistics and probability theory are thus clearly emphasized from the very beginning and, moreover, certain moment problems and probability approximation schemes can be treated quite naturally within the scope of the book.

The text requires that the reader is familiar with probability and statistics and with some mathematical programming. To make the book easily accessible for probabilists and statisticians, the first three chapters present an extensive introduction to the theory and algorithms for linear programming. Background material on log-concave and quasi- concave measures is included in Chapter 4 and an appendix summarizes properties of multivariate normal distributions.

Special attention should be paid to the following three chapters. Chapter 5 provides an extensive exposition on moment problems with optimization including discrete moment problems; these are stochastic programming problems in the light of the extended definition and they can be efficiently exploited in bounding procedures and approximations for other stochastic programs. Various problems of bounding and approximation of probabilities are investigated in Chapter 6. Chapter 7 is concerned with principles of statistical decisions and it gives a background for design of the stochastic programming models treated in the subsequent chapters.

The rest of the book centres around theory, applications and algorithms for the common stochastic programming problems: the survey on various static stochastic programming models (Chapter 8); the presentation of results on the solution of the simple recourse problems (Chapter 9), on mathematical properties of two-stage stochastic programs and on selected techniques of their solution (Chapter 12); the comprehensive chapters on the convexity properties of probabilistic constrained problems (Chapter 10) and on experience in their numerical treatment (Chapter 11); discussion of the basic ideas about multistage stochastic programs (Chapter 13). Chapter 14, based mostly on the personal experience of the author, gives an insight into many interesting application areas. Distribution problems and various asymptotic results for random linear programs are the subject of Chapter 15.

For the most part, the chapters include material both on the theory and applications. There is a broad coverage of various solution techniques which are described and discussed without details about appropriate computer codes for implementing them. A fair number of examples is scattered throughout the text and the Exercises and Problems sections present an original selection.

With the reference list of 538 entries it is clear that there has been a large amount of research activity in the field with a significant contribution of the author. The monograph provides by far the broadest discussion of stochastic programming currently available with new insights namely from the point of view of probability and statistics. It is partly written at the level of advanced research papers and is primarily intended both for researchers in operations research and for mathematicians and statisticians. The author recommends is also a textbook for advanced graduate and postgraduate students. Nevertheless, numerous described case studies and real life examples give a guarantee that the book will attract the attention of a broader group of researchers coming from economics, finance, engineering, etc.

The text requires that the reader is familiar with probability and statistics and with some mathematical programming. To make the book easily accessible for probabilists and statisticians, the first three chapters present an extensive introduction to the theory and algorithms for linear programming. Background material on log-concave and quasi- concave measures is included in Chapter 4 and an appendix summarizes properties of multivariate normal distributions.

Special attention should be paid to the following three chapters. Chapter 5 provides an extensive exposition on moment problems with optimization including discrete moment problems; these are stochastic programming problems in the light of the extended definition and they can be efficiently exploited in bounding procedures and approximations for other stochastic programs. Various problems of bounding and approximation of probabilities are investigated in Chapter 6. Chapter 7 is concerned with principles of statistical decisions and it gives a background for design of the stochastic programming models treated in the subsequent chapters.

The rest of the book centres around theory, applications and algorithms for the common stochastic programming problems: the survey on various static stochastic programming models (Chapter 8); the presentation of results on the solution of the simple recourse problems (Chapter 9), on mathematical properties of two-stage stochastic programs and on selected techniques of their solution (Chapter 12); the comprehensive chapters on the convexity properties of probabilistic constrained problems (Chapter 10) and on experience in their numerical treatment (Chapter 11); discussion of the basic ideas about multistage stochastic programs (Chapter 13). Chapter 14, based mostly on the personal experience of the author, gives an insight into many interesting application areas. Distribution problems and various asymptotic results for random linear programs are the subject of Chapter 15.

For the most part, the chapters include material both on the theory and applications. There is a broad coverage of various solution techniques which are described and discussed without details about appropriate computer codes for implementing them. A fair number of examples is scattered throughout the text and the Exercises and Problems sections present an original selection.

With the reference list of 538 entries it is clear that there has been a large amount of research activity in the field with a significant contribution of the author. The monograph provides by far the broadest discussion of stochastic programming currently available with new insights namely from the point of view of probability and statistics. It is partly written at the level of advanced research papers and is primarily intended both for researchers in operations research and for mathematicians and statisticians. The author recommends is also a textbook for advanced graduate and postgraduate students. Nevertheless, numerous described case studies and real life examples give a guarantee that the book will attract the attention of a broader group of researchers coming from economics, finance, engineering, etc.

Reviewer: J.Dupačová (Praha)

##### MSC:

90C15 | Stochastic programming |

90C05 | Linear programming |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

65C99 | Probabilistic methods, stochastic differential equations |