LaTeX
\nabla f(x)=\lambda\nabla g(x)Variables
- f: objective
- g: equality constraint
- λ: Lagrange multiplier
How to use this formula
States the first-order condition for an equality-constrained extremum with one regular constraint.
Important notes
- Constraint qualification is required.
- Multiple constraints use a sum of multiplier-weighted gradients.
Quick example
Optimize f subject to g=0 by solving ∇f=λ∇g together with g=0.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Constraint qualification is required.
- Specify the objective, constraints, feasible domain, and any convexity or differentiability assumptions used to justify the result.
Worked example
Optimize f subject to g=0 by solving ∇f=λ∇g together with g=0.
Common mistakes
- Before substituting values into Lagrange Multiplier Condition, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- Do not treat a stationary point as a global optimum without checking constraints, boundaries, and the required optimality conditions.
Continue the workflow
Use Lagrange Multiplier Condition in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\nabla f(x)=\lambda\nabla g(x). - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Lagrange Multiplier Condition used for?
States the first-order condition for an equality-constrained extremum with one regular constraint.
Can I copy this formula as LaTeX?
Yes. Copy \nabla f(x)=\lambda\nabla g(x) or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.