LaTeX
\nabla f(x)+\sum_i\lambda_i\nabla g_i(x)+\sum_j\nu_j\nabla h_j(x)=0Variables
- f: objective
- g_i: inequality constraints
- h_j: equality constraints
- λ_i,ν_j: multipliers
How to use this formula
Gives the stationarity part of the Karush–Kuhn–Tucker conditions.
Important notes
- Full KKT conditions also include feasibility and complementary slackness.
- For convex problems under regularity, KKT conditions can be sufficient.
Quick example
Combine stationarity with λ_i≥0 and λ_i g_i(x)=0.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Full KKT conditions also include feasibility and complementary slackness.
- For the KKT Stationarity, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.
- Specify the objective, constraints, feasible domain, and any convexity or differentiability assumptions used to justify the result.
Worked example
Combine stationarity with λ_i≥0 and λ_i g_i(x)=0.
Common mistakes
- Do not omit the index or bounds in KKT Stationarity; changing either one changes which terms are included.
- Do not treat a stationary point as a global optimum without checking constraints, boundaries, and the required optimality conditions.
Continue the workflow
Use KKT Stationarity 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)+\sum_i\lambda_i\nabla g_i(x)+\sum_j\nu_j\nabla h_j(x)=0. - 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 KKT Stationarity used for?
Gives the stationarity part of the Karush–Kuhn–Tucker conditions.
Can I copy this formula as LaTeX?
Yes. Copy \nabla f(x)+\sum_i\lambda_i\nabla g_i(x)+\sum_j\nu_j\nabla h_j(x)=0 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.