# Introduction

Linear programs (LPs) are a fundamental class of optimization problems of the form:

\begin{align} \min_{x \in \mathbb{R}^n} & \sum\limits_{i=1}^n c_i x_i \\ \;\;\text{s.t.} & l_j \le \sum\limits_{i=1}^n a_{ij} x_i \le u_j & j = 1 \ldots m \\ & l_i \le x_i \le u_i & i = 1 \ldots n. \end{align}

The most important thing to note is that all terms are of the form coefficient * variable, and that there are no nonlinear terms or multiplications between variables.

Mixed-integer linear programs (MILPs) are extensions of linear programs in which some (or all) of the decision variables take discrete values.

## How to choose a solver

Almost all solvers support linear programs; look for "LP" in the list of Supported solvers. However, fewer solvers support mixed-integer linear programs. Solvers supporting discrete variables start with "(MI)" in the list of Supported solvers.

## How these tutorials are structured

Having a high-level overview of how this part of the documentation is structured will help you know where to look for certain things.

• The following tutorials are worked examples that present a problem in words, then formulate it in mathematics, and then solve it in JuMP. This usually involves some sort of visualization of the solution. Start here if you are new to JuMP.
• The Tips and tricks tutorial contains a number of helpful reformulations and tricks you can use when modeling linear programs. Look here if you are stuck trying to formulate a problem as a linear program.
• The Sensitivity analysis of a linear program tutorial explains how to create sensitivity reports like those produced by the Excel Solver.
• The Callbacks tutorial explains how to write a variety of solver-independent callbacks. Look here if you want to write a callback.
• The remaining tutorials are less verbose and styled in the form of short code examples. These tutorials have less explanation, but may contain useful code snippets, particularly if they are similar to a problem you are trying to solve.