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 ${\displaystyle P}$ and a vector ${\displaystyle \mathbf {x} ^{*}\in \mathbb {R} ^{n}}$, ${\displaystyle \mathbf {x} ^{*}}$ is a basic solution if:

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

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

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

References

1. Bertsimas, Dimitris; Tsitsiklis, John N. (1997). Introduction to linear optimization. Belmont, Mass.: Athena Scientific. p. 50. ISBN 978-1-886529-19-9.