The book belongs to the field of stochastic programming that arose in the fifties and sixties. Namely, at this time it was recognised that the random character of some parameters in optimization models cannot be neglected. As a result of this fact a new field of mathematical programming theory, called “stochastic programming”, arose and has developed rather quickly as an independent part of mathematical programming. The monograph under review is devoted to special types of stochastic programming problems. In particular it is devoted to the problems with probability and quantile objective. It is known that (under rather general assumptions) the corresponding objective functions are close to each other. Both authors are specialists in the above mentioned directions of stochastic programming.
The book is divided into 4 chapter. Examples of applications leading to the investigated types of stochastic problems are (mainly) introduced in chapter one. In the second chapter we can find the necessary mathematical tools for the investigation of the problems. Mathematical properties of the problems are investigated there. We can find (in this chapter) not only the investigation of the convexity and continuity but also existence conditions for differentiability of the objective functionals. Moreover, the forms of gradient vectors are presented. The case when the underlying probability measure is unknown is investigated in chapter 3. To this end kernel estimates, bounds of probability as well as properties of quasi-convex functions are recalled. The last chapter is devoted to the methods and algorithms for solving the problems. The relationship between the solutions of the two investigated problems can also be found in this chapter. The chapters are completed by individual lists of references
According to the knowledge of the reviewer this is the first monograph dealing completely with these special types of stochastic programming problems.