﻿

# Lagrange multipliers on Banach spaces

In the field of calculus of variations in mathematics, the method of Lagrange multipliers on Banach spaces can be used to solve certain infinite-dimensional constrained optimization problems. The method is a generalization of the classical method of Lagrange multipliers as used to find extrema of a function of finitely many variables.

The Lagrange multiplier theorem for Banach spaces

Let "X" and "Y" be real Banach spaces. Let "U" be an open subset of "X" and let "f" : "U" → R be a continuously differentiable function. Let "g" : "U" → "Y" be another continuously differentiable function, the "constraint": the objective is to find the extremal points (maxima or minima) of "f" subject to the constraint that "g" is zero.

Suppose that "u"0 is a "constrained extremum" of "f", i.e. an extremum of "f" on

:$g^\left\{-1\right\} \left(0\right) = \left\{ x in U | g\left(x\right) = 0 in Y \right\} subseteq U.$

Suppose also that the Fréchet derivative D"f"("u"0) : "X" → R of "f" at "u"0 is a surjective linear map. Then there exists a Lagrange multiplier "λ" : "Y" → R in "Y", the dual space to "Y", such that

:$mathrm\left\{D\right\} f \left(u_\left\{0\right\}\right) = lambda circ mathrm\left\{D\right\} g \left(u_\left\{0\right\}\right). quad mbox\left\{\left(L\right)\right\}$

Since D"f"("u"0) is an element of the dual space "X", equation (L) can also be written as

:$mathrm\left\{D\right\} f \left(u_\left\{0\right\}\right) = left\left( mathrm\left\{D\right\} g \left(u_\left\{0\right\}\right) ight\right)^\left\{*\right\} \left(lambda\right),$

where (D"g"("u"0))("λ") is the pullback of "λ" by D"g"("u"0), i.e. the action of the adjoint map (D"g"("u"0)) on "λ", as defined by

:$left\left( mathrm\left\{D\right\} g \left(u_\left\{0\right\}\right) ight\right)^\left\{*\right\} \left(lambda\right) = lambda circ mathrm\left\{D\right\} g \left(u_\left\{0\right\}\right).$

Connection to the finite-dimensional case

In the case that "X" and "Y" are both finite-dimensional (i.e. linearly isomorphic to R"m" and R"n" for some natural numbers "m" and "n") then writing out equation (L) in matrix form shows that "λ" is the usual Lagrange multiplier vector; in the case "m" = "n" = 1, "λ" is the usual Lagrange multiplier, a real number.

Application

In many optimization problems, one seeks to minimize a functional defined on an infinite-dimensional space such as a Banach space.

Consider, for example, the Sobolev space "X" = "H"01( [−1, +1] ; R) and the functional "f" : "X" → R given by

:$f\left(u\right) = int_\left\{-1\right\}^\left\{+1\right\} u\text{'}\left(x\right)^\left\{2\right\} , mathrm\left\{d\right\} x.$

Without any constraint, the minimum value of "f" would be 0, attained by "u"0("x") = 0 for all "x" between −1 and +1. One could also consider the constrained optimization problem, to minimize "f" among all those "u" ∈ "X" such that the mean value of "u" is +1. In terms of the above theorem, the constraint "g" would be given by

:$g\left(u\right) = frac\left\{1\right\}\left\{2\right\} int_\left\{-1\right\}^\left\{+1\right\} u\left(x\right) , mathrm\left\{d\right\} x - 1.$

The method of Lagrange multipliers on Banach spaces is required in order to solve this problem.

References

*

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Lagrange multipliers — In mathematical optimization problems, the method of Lagrange multipliers, named after Joseph Louis Lagrange, is a method for finding the extrema of a function of several variables subject to one or more constraints; it is the basic tool in… …   Wikipedia

• Lagrange multiplier — Figure 1: Find x and y to maximize f(x,y) subject to a constraint (shown in red) g(x,y) = c …   Wikipedia

• List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia

• List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra …   Wikipedia

• Pontryagin's minimum principle — Pontryagin s maximum (or minimum) principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It …   Wikipedia

• List of important publications in mathematics — One of the oldest surviving fragments of Euclid s Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5. This is a list of important publications in mathematics, organized by field. Some… …   Wikipedia

• Singular value decomposition — Visualization of the SVD of a 2 dimensional, real shearing matrix M. First, we see the unit disc in blue together with the two canonical unit vectors. We then see the action of M, which distorts the disk to an ellipse. The SVD decomposes M into… …   Wikipedia

• Compact operator on Hilbert space — In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite rank operators in the uniform operator topology. As such, results from matrix theory… …   Wikipedia

• Mathematical economics — Economics …   Wikipedia