# 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 -- linear functions. A **linear optimization (LO)** is an optimization problem of the form

$$
\begin{align*}
    \min \quad & c^\top x \\
    \text{s.t.} \quad & A x \leq b\\
    & x \geq 0,
\end{align*}
$$

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 $x_i$'s. In fact, any LO problem with such features can be easily converted to the "canonical" 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](bim.ipynb)
* [Least Absolute Deviation (LAD) Regression](lad-regression.ipynb)
* [Mean Absolute Deviation (MAD) portfolio optimization](mad-portfolio-optimization.ipynb)
* [Wine quality prediction problem using L1 regression](L1-regression-wine-quality.ipynb)
* [A variant of BIM problem: maximizing the lowest possible profit](bim-maxmin.ipynb)
* [Two variants of the BIM problem using fractional objective or additional fixed costs](bim-fractional.ipynb)
* [The BIM production problem using demand forecasts](bim-rawmaterialplanning)
* [Extra material: Multi-product facility production](multiproductionfaciliity_worstcase.ipynb)


Go to the [next chapter](../03/03.00.md) about mixed-integer linear optimization.