Introduction

This chapter provides an introduction to the vibrant field of stochastic programming (SP), which is a branch of mathematical programming that deals with optimization problems involving data uncertainties and risk. We begin with the motivation and explain why SP has become so pervasive in operations research, science, and engineering and discuss some of its diverse set of example applications that span our everyday lives. In Sect. 1.2, we provide preliminaries needed for topics covered in later chapters. We start with defining the basic notation used throughout the book and review selected fundamental concepts from both convexity theory and probability and statistics. These concepts include convex sets and convex functions, separation hyperplanes, random variables, and probability spaces. In Sect. 1.3, we provide a roadmap from the deterministic setting of linear programming (LP) to the stochastic setting of SP. We assume that the reader is familiar with LP and skip all the fundamental concepts of LP such as duality theory and sensitivity analysis. Understanding these LP concepts is important to studying SP. We first introduce scenario trees for representing the underlying stochastic data process for a given SP problem and then introduce several example approaches for dealing with uncertainty and risk. These approaches include the expected value solution, scenario analysis, two-stage recourse model, probabilistic (chance) constraints model, integrated-chance constraints, and multistage recourse model. We end the chapter with some bibliographic notes in Sect. 1.3.9