# 2. Linear Optimization#

The simplest and most scalable class of optimization problems is the one where the objective function and the constraints are formulated using the simplest possible type of functions â linear functions. A **linear optimization (LO)** is an optimization problem of the form

where the \(n\) (decision) variables are grouped in a vector \(x \in \mathbb{R}^n\), \(c \in \mathbb{R}^n\) are the objective coefficients, and the \(m\) linear constraints are described by the matrix \(A \in \mathbb{R}^{m \times n}\) and the vector \(b \in \mathbb{R}^m\).

Linear problems can (i) be maximization problems, (ii) involve equality constraints and constraints of the form \(\geq\), and (iii) have unbounded or non-positive decision variables \(x_i\)âs. In fact, any LO problem with such features can be easily converted to the âcanonicalâ LO form above by adding/removing variables and/or multiplying specific inequalities by \(-1\).

In this chapter, there is a number of examples with companion AMPL implementation that explore various modeling and implementation aspects of LOs:

A first LO example, modelling the microchip production problem of company BIM

A variant of BIM problem: maximizing the lowest possible profit

Two variants of the BIM problem using fractional objective or additional fixed costs

Go to the next chapter about mixed-integer linear optimization.