Lagrange multipliers on Banach spaces
Let $U$ be open in a real Banach space^{} $X$, and $Y$ be another real Banach space. Let $f:U\to \mathbb{R}$ and $g:U\to Y$ be continuously differentiable functions.
Suppose that $a$ is a minimum or maximum point of $f$ on $M=\{x\in U:g(x)=0\}$, and the Fréchet derivative $\mathrm{D}g(a):X\to Y$ is surjective. Then there exists a Lagrange multiplier^{} vector $\lambda \in {Y}^{*}$ such that
$$\mathrm{D}f(a)=\mathrm{D}g{(a)}^{*}\lambda =\lambda \circ \mathrm{D}g(a).$$ |
(The function $\mathrm{D}g{(a)}^{*}:{Y}^{*}\to {X}^{*}$ denotes the pullback or adjoint^{} by $\mathrm{D}g(a)$ on the continuous duals, defined by the second equality.)
If $X$ and $Y$ 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.
References
- 1 Eberhard Zeidler. Applied functional analysis^{}: main principles and their applications. Springer-Verlag, 1995.
Title | Lagrange multipliers on Banach spaces |
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Canonical name | LagrangeMultipliersOnBanachSpaces |
Date of creation | 2013-03-22 15:28:30 |
Last modified on | 2013-03-22 15:28:30 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 5 |
Author | stevecheng (10074) |
Entry type | Theorem |
Classification | msc 49-00 |
Classification | msc 49K35 |