1 - Introduction

Summary

Optimization is a branch of applied mathematics that derives its importance both from the wide variety of its applications and from the availability of efficient algorithms. Mathematically, it refers to the minimization (or maximization) of a given objective function of several decision variables that satisfy functional constraints. A typical optimization model addresses the allocation of scarce resources among possible alternative uses in order to maximize an objective function such as total profit.

Decision variables, the objective function, and constraints are three essential elements of any optimization problem. Problems that lack constraints are called unconstrained optimization problems, while others are often referred to as constrained optimization problems. Problems with no objective functions are called feasibility problems. Some problems may have multiple objective functions. These problems are often addressed by reducing them to a single-objective optimization problem or a sequence of such problems.

If the decision variables in an optimization problem are restricted to integers, or to a discrete set of possibilities, we have an integer or discrete optimization problem. If there are no such restrictions on the variables, the problem is a continuous optimization problem. Of course, some problems may have a mixture of discrete and continuous variables. We continue with a list of problem classes that we will encounter in this book.