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.