Lectures on Stochastic Programming

Excerpt

The main topic of this book is optimization problems involving uncertain parameters, for which stochastic models are available. Although many ways have been proposed to model uncertain quantities, stochastic models have proved their flexibility and usefulness in diverse areas of science. This is mainly due to solid mathematical foundations and theoretical richness of the theory of probability and stochastic processes, and to sound statistical techniques of using real data.

Optimization problems involving stochastic models occur in almost all areas of science and engineering, from telecommunication and medicine to finance. This stimulates interest in rigorous ways of formulating, analyzing, and solving such problems. Due to the presence of random parameters in the model, the theory combines concepts of the optimization theory, the theory of probability and statistics, and functional analysis. Moreover, in recent years the theory and methods of stochastic programming have undergone major advances. All these factors motivated us to present in an accessible and rigorous form contemporary models and ideas of stochastic programming. We hope that the book will encourage other researchers to apply stochastic programming models and to undertake further studies of this fascinating and rapidly developing area.

We do not try to provide a comprehensive presentation of all aspects of stochastic programming, but we rather concentrate on theoretical foundations and recent advances in selected areas. The book is organized into seven chapters. The first chapter addresses modeling issues. The basic concepts, such as recourse actions, chance (probabilistic) constraints, and the nonanticipativity principle, are introduced in the context of specific models. The discussion is aimed at providing motivation for the theoretical developments in the book, rather than practical recommendations.

Chapters 2 and 3 present detailed development of the theory of two-stage and multistage stochastic programming problems. We analyze properties of the models and develop optimality conditions and duality theory in a rather general setting. Our analysis covers general distributions of uncertain parameters and provides special results for discrete distributions, which are relevant for numerical methods. Due to specific properties of two- and multistage stochastic programming problems, we were able to derive many of these results without resorting to methods of functional analysis.