Karush–Kuhn–Tucker conditions
E83405
mathematical concept
necessary conditions for optimality
optimality conditions
result in nonlinear programming
The Karush–Kuhn–Tucker conditions are fundamental optimality criteria in nonlinear programming that generalize Lagrange multipliers to handle inequality constraints.
Observed surface forms (2)
| Surface form | As subject | As object |
|---|---|---|
| Kuhn–Tucker conditions | 0 | 2 |
| Karush–Kuhn–Tucker conditions in nonlinear programming | 0 | 1 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
→
necessary conditions for optimality → optimality conditions → result in nonlinear programming → |
| alsoKnownAs | KKT conditions → |
| appliesTo |
constrained optimization problems
→
nonlinear programming problems → optimization problems with inequality constraints → |
| are |
necessary conditions for optimality under suitable constraint qualifications
→
sufficient conditions for optimality in convex optimization problems → |
| assumes | constraint qualification such as Slater’s condition in convex problems → |
| category |
Mathematical optimization theorems
→
Nonlinear programming → |
| component |
complementary slackness condition
→
constraint qualification assumption → dual feasibility condition → primal feasibility condition → stationarity condition → |
| expressedAs | system of equations and inequalities → |
| field |
mathematical optimization
→
nonlinear programming → optimization theory → |
| formalizedIn | Lagrangian saddle-point framework → |
| generalizes | method of Lagrange multipliers → |
| historicalOrigin |
independent work of Kuhn and Tucker in the 1950s
→
work of William Karush in 1939 → |
| implies | zero product between each inequality constraint and its multiplier at optimum → |
| involves |
Lagrange multipliers for equality constraints
→
Lagrange multipliers for inequality constraints → Lagrangian function → |
| namedAfter |
Albert W. Tucker
→
Harold W. Kuhn → William Karush → |
| relatedTo |
Fritz John
→
surface form:
Fritz John conditions
Lagrange multipliers →
surface form:
Lagrange duality
first-order necessary conditions → |
| relates | primal variables to Lagrange multipliers → |
| requires |
differentiability of objective and constraint functions in standard form
→
nonnegativity of multipliers for inequality constraints → |
| usedFor |
analyzing sensitivity in optimization
→
characterizing local optima → deriving dual problems → |
| usedIn |
convex optimization
→
economics → engineering design optimization → machine learning → operations research → support vector machines → |
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Kuhn–Tucker conditions
this entity surface form:
Kuhn–Tucker conditions
this entity surface form:
Karush–Kuhn–Tucker conditions in nonlinear programming