Math 2.0

Operations research, mathematical programming Theorem

Lagrange multipliers

Builds on Constrained optimisationLocal maximum and minimumGradientJacobian matrixImplicit function theoremC^k function characterised by continuous derivatives

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, let m<nm < n, and let f ⁣:U→ℝf \colon U \to \mathbb{R} and g=(g1,…,gm) ⁣:U→ℝmg = (g_1, \dots, g_m) \colon U \to \mathbb{R}^m be of class C^1. Let M={x∈U:g(x)=0}M = \{x \in U : g(x) = 0\}, and let a∈Ma \in M be a local extremum of the restriction of ff to MM. Suppose that the gradients ∇g1(a),…,∇gm(a)\nabla g_1(a), \dots, \nabla g_m(a) are linearly independent. Then there is a unique λ=(λ1,…,λm)∈ℝm\lambda = (\lambda_1, \dots, \lambda_m) \in \mathbb{R}^m such that ∇f(a)=λ1∇g1(a)+⋯+λm∇gm(a).\nabla f(a) = \lambda_1 \nabla g_1(a) + \dots + \lambda_m \nabla g_m(a).

Remarks

The numbers λi\lambda_i are the Lagrange multipliers. Equivalently, (a,λ)(a, \lambda) is a critical point of the Lagrangian L(x,λ)=f(x)−λ⋅g(x)L(x, \lambda) = f(x) - \lambda \cdot g(x). The hypothesis on the gradients is the constraint qualification. It says that the Jacobian matrix Dg(a)Dg(a) has rank mm. The proof uses the implicit function theorem. Near aa, the set MM is the graph of a C1C^1 function of n−mn - m variables. The gradient of ff is orthogonal to the tangent space of MM at aa. The condition is necessary, not sufficient. Each candidate point needs a further check, for example a comparison of values on a compact set MM.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptLocal maximum and minimum, definition, no evidence levelLocal maximum and min…definitionLocal maximum and minimum, concept, no evidence levelLocal maximum and min…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionConstrained optimisation, concept, no evidence levelConstrained optimisat…conceptDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionChain rule in several variables, statement, no evidence levelChain rule in several…statementGradient, definition, no evidence levelGradientdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionIterated derivative, definition, no evidence levelIterated derivativedefinitionDerivative of the inverse function, statement, no evidence levelDerivative of the inv…statementJacobian determinant, definition, no evidence levelJacobian determinantdefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementInverse function theorem, statement, no evidence levelInverse function theo…statementImplicit function theorem, statement, no evidence levelImplicit function the…statementLagrange multipliers, statement, no evidence levelLagrange multipliersstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Euclidean space object
  2. Function concept
  3. Function composition concept
  4. Fréchet derivative definition
  5. Function of several variables concept
  6. Limit of a function of several variables concept
  7. Local maximum and minimum definition
  8. Local maximum and minimum concept
  9. Partial derivative concept
  10. Differentiable function of several variables definition
  11. Partial derivative definition
  12. Constrained optimisation concept
  13. Differentiable on an open set definition
  14. Chain rule in several variables statement
  15. Gradient definition
  16. Higher partial derivatives definition
  17. Jacobian matrix definition
  18. Iterated derivative definition
  19. Derivative of the inverse function statement
  20. Jacobian determinant definition
  21. C^k function characterised by continuous derivatives statement
  22. Inverse function theorem statement
  23. Implicit function theorem statement
  24. Lagrange multipliers statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Lagrange_multiplier, Statement
  2. Web https://en.wikipedia.org/wiki/Lagrange_multiplier, Multiple constraints
About this page
Tag0VKKP8X6Q6
URIm2://0VKKP8X6Q6
Aliaseslagrange-multiplier-theorem, method-of-lagrange-multipliers
Kindstatement, theorem
WritingW1 stated
MSC90C46
StatusThis copy of the site has no build data. Run m2 status 0VKKP8X6Q6.
UsesConstrained optimisation, Local maximum and minimum, Gradient, Jacobian matrix, Implicit function theorem, C^k function characterised by continuous derivatives
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