# Lagrange multiplier rule

### From Glossary

From the extension of Lagrange's multiplier theorem. Suppose

where , , and are smooth. Then, there exist multipliers for which the following conditions hold:

- ;
- ;
- .

Since the last condition, given , is equivalent to complementary slackness. These are considered first-order optimality conditions, though the Lagrange Multiplier Rule is not always valid -- see constraint qualifications.

For extensions see the Generalized Lagrange multiplier method.