Basic solution (linear programming)

In linear programming, a discipline within applied mathematics, a basic solution is any solution of a linear programming problem satisfying certain specified technical conditions. For a polyhedron $P$ and a vector ${\mathbf{𝐱}}^{*}\in {\mathbb{ℝ}}^n$, ${\mathbf{𝐱}}^{*}$ is a basic solution if:

1. All the equality constraints defining $P$ are active at ${\mathbf{𝐱}}^{*}$
2. Of all the constraints that are active at that vector, at least $n$ of them must be linearly independent. Note that this also means that at least $n$ constraints must be active at that vector.

A constraint is active for a particular solution $\mathbf{𝐱}$ if it is satisfied at equality for that solution.

A basic solution that satisfies all the constraints defining $P$ (or, in other words, one that lies within $P$) is called a basic feasible solution.