PlanetMath: Lagrange multipliers on Banach spaces

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Lagrange multipliers on Banach spaces

(Theorem)

Let be open in a real Banach space , and be another real Banach space. Let $f\colon U \to \mathbb{R}$ and $g\colon U \to Y$ be continuously differentiable functions.

Suppose that is a minimum or maximum point of on $M = \{ x \in U : g(x) = 0 \}$ , and the Fréchet derivative ${\mathrm{D}}g(a)\colon X \to Y$ is surjective. Then there exists a Lagrange multiplier vector $\lambda \in Y^*$ such that

$\displaystyle {\mathrm{D}}f(a) = {\mathrm{D}}g(a)^* \lambda = \lambda \circ {\mathrm{D}}g(a)\,.$

(The function ${\mathrm{D}}g(a)^*\colon Y^* \to X^*$ denotes the pullback or adjoint by ${\mathrm{D}}g(a)$ on the continuous duals, defined by the second equality.)

If and are finite-dimensional, writing out the above equation in matrix form shows that $\lambda$ really is the usual Lagrange multiplier vector. The condition that ${\mathrm{D}}g(a)$ is surjective means that ${\mathrm{D}}g(a)$ must have full rank as a matrix.

Bibliography

1: Eberhard Zeidler. Applied functional analysis: main principles and their applications. Springer-Verlag, 1995.

"Lagrange multipliers on Banach spaces" is owned by stevecheng.

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Cross-references: rank, matrix, equation, finite-dimensional, equality, continuous duals, adjoint, pullback, vector, surjective, Fréchet derivative, point, functions, continuously differentiable, Banach space, real, open

This is version 2 of Lagrange multipliers on Banach spaces, born on 2005-08-16, modified 2005-08-17.
Object id is 7329, canonical name is LagrangeMultipliersInBanachSpaces.
Accessed 1802 times total.

Classification:

AMS MSC:	49K35 (Calculus of variations and optimal control; optimization :: Necessary conditions and sufficient conditions for optimality :: Minimax problems)
	49-00 (Calculus of variations and optimal control; optimization :: General reference works )

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