This chapter introduces some basic mathematical notions that are used throughout the book. Convex sets and functions, optimization problems, feasible solutions, and optimal solutions are first defined. The chapter then covers duality theory, including the definition of the Lagrangian function and the dual function, which are used to derive the duals of linear programs. Weak and strong duality are then defined and related to certain classes of optimization problems. The Karush–Kuhn–Tucker (KKT) conditions are defined, and their relation to the optimal solution of mathematical programs is discussed. KKT conditions are a fundamental concept used extensively in the book in order to understand the properties and economic interpretations of the various economic models encountered. Subgradients are subsequently defined in order to establish the relation between Lagrange multipliers and the sensitivity of an optimization model with respect to changes in the right-hand side parameters of its constraints. These sensitivity results are also used repeatedly in the book, for instance in order to derive locational marginal prices in chapter 5.

Duality theory has a central role in constrained optimization, both from a theoretical point of view and to enable understanding of solution methods and problem reformulations for special classes of problems.Such applications are presented in the next chapter on Lagrangian relaxation and Lagrangian decomposition.In this chapter, the fundamental background for duality theory is presented along with a basic introduction of key concepts related to it.

This is one of the main and key chapters in the introductory material part of this book.Constrained nonlinear programming, involving both equality and inequality constraints, is introduced and related in an intuitive (at this stage) manner with Lagrange multipliers.In a later chapter (duality theory, Chapter 17) a more rigorous and theoretical introduction to Lagrangian theory is presented.