Math 2.0

Operations research, mathematical programming Corollary

Lagrange multiplier with one constraint

Builds on Lagrange multipliersGradientLevel setGradient is orthogonal to level sets

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open with n≥2n \ge 2, and let f,g ⁣:U→ℝf, g \colon U \to \mathbb{R} be of class C1C^1. Let c∈ℝc \in \mathbb{R}, and let aa be a point of the level set {x∈U:g(x)=c}\{x \in U : g(x) = c\} with ∇g(a)≠0\nabla g(a) \ne 0. If the restriction of ff to this level set has a local extremum at aa, then there is a unique λ∈ℝ\lambda \in \mathbb{R} with ∇f(a)=λ ∇g(a).\nabla f(a) = \lambda \, \nabla g(a).

Remarks

This is the case m=1m = 1 of the Lagrange multiplier theorem, applied to g−cg - c. Geometrically, the level set of ff through aa is tangent to the constraint set at aa, because both gradients are orthogonal to it. Example: to maximise f(x,y)=x+yf(x, y) = x + y on x2+y2=1x^2 + y^2 = 1, solve (1,1)=λ(2x,2y)(1, 1) = \lambda (2 x, 2 y) and x2+y2=1x^2 + y^2 = 1. The maximum is 2\sqrt{2}, at (1/2,1/2)(1/\sqrt{2}, 1/\sqrt{2}). Without the condition ∇g(a)≠0\nabla g(a) \ne 0 the conclusion can fail. For g(x,y)=y2−x3g(x, y) = y^2 - x^3 and f(x,y)=xf(x, y) = x, the minimum on g=0g = 0 is at (0,0)(0, 0), where ∇g=0\nabla g = 0 and ∇f=(1,0)\nabla f = (1, 0).

Reading path

To understand this page from the ground up, read these 34 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptPreimage of a set, concept, no evidence levelPreimage of a setconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptParametrised curve, concept, no evidence levelParametrised curveconceptLocal 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 derivativeconceptLevel set, definition, no evidence levelLevel setdefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionConstrained optimisation, concept, no evidence levelConstrained optimisat…conceptGradient, concept, no evidence levelGradientconceptDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionChain rule in several variables, statement, no evidence levelChain rule in several…statementHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionGradient, definition, no evidence levelGradientdefinitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionIterated derivative, definition, no evidence levelIterated derivativedefinitionChain rule for a composition with a curve, statement, no evidence levelChain rule for a comp…statementDerivative 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…statementGradient is orthogonal to level sets, statement, no evidence levelGradient is orthogona…statementInverse function theorem, statement, no evidence levelInverse function theo…statementImplicit function theorem, statement, no evidence levelImplicit function the…statementLagrange multipliers, statement, no evidence levelLagrange multipliersstatementLagrange multiplier with one constraint, statement, no evidence levelLagrange multiplier w…statement
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.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Function composition concept
  4. Preimage of a set concept
  5. Fréchet derivative definition
  6. Total derivative concept
  7. Function of several variables concept
  8. Vector-valued function concept
  9. Limit of a function of several variables concept
  10. Differentiable function of several variables concept
  11. Parametrised curve concept
  12. Local maximum and minimum definition
  13. Local maximum and minimum concept
  14. Partial derivative concept
  15. Level set definition
  16. Differentiable function of several variables definition
  17. Partial derivative definition
  18. Constrained optimisation concept
  19. Gradient concept
  20. Differentiable on an open set definition
  21. Chain rule in several variables statement
  22. Higher partial derivatives definition
  23. Gradient definition
  24. Jacobian matrix definition
  25. Iterated derivative definition
  26. Chain rule for a composition with a curve statement
  27. Derivative of the inverse function statement
  28. Jacobian determinant definition
  29. C^k function characterised by continuous derivatives statement
  30. Gradient is orthogonal to level sets statement
  31. Inverse function theorem statement
  32. Implicit function theorem statement
  33. Lagrange multipliers statement
  34. Lagrange multiplier with one constraint statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Lagrange_multiplier, Single constraint
About this page
Tag04A2XEX4D4
URIm2://04A2XEX4D4
Aliaseslagrange-multiplier-single-constraint
Kindstatement, corollary
WritingW0 stub
MSC90C46
StatusThis copy of the site has no build data. Run m2 status 04A2XEX4D4.
UsesLagrange multipliers, Gradient, Level set, Gradient is orthogonal to level sets

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