# Lagrange multipliers

(1.3 hours to learn)

## Summary

Lagrange multipliers are a tool for solving optimization problems with equality or inequality constraints. In particular, some linear combination of the gradients of the constraints must match the gradient of the function. Lagrange multipliers are a key idea behind Lagrange duality, a central concept in convex optimization.

## Context

This concept has the prerequisites:

## Core resources (read/watch one of the following)

## -Paid-

→ Multivariable Calculus

An introductory multivariable calculus textbook.

Location:
Section 13.9, "Lagrange multipliers and constrained optimization," pages 918-924

Other notes:

- Don't worry about the proofs as far as this node is concerned.

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Section 5.4, "Lagrange multipliers," pages 216-222

Other notes:

- Don't worry about the proofs as far as this node is concerned.

## Supplemental resources (the following are optional, but you may find them useful)

## -Free-

→ MIT Open Courseware: Multivariable Caclulus (2010)

Video lectures for MIT's introductory multivariable calculus class.

→ Lagrange multipliers without permanent scarring

## See also

- We can't always find a closed form solution to an optimization problem. Here are some computational techniques we can use: Some widely used classes of optimization problems: