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 u₀ is a constrained extremum of f, i.e. an extremum of f on

$g^{-1}(0)=\{x\in U\mid g(x)=0\in Y\}\subseteq U.$

Suppose also that the Fréchet derivative Dg(u₀) : X → Y of g at u₀ is a surjective linear map. Then there exists a Lagrange multiplier λ : Y → R in Y^∗, the dual space to Y, such that

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

Since Df(u₀) is an element of the dual space X^∗, equation (L) can also be written as

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

where (Dg(u₀))^∗(λ) is the pullback of λ by Dg(u₀), i.e. the action of the adjoint map (Dg(u₀))^∗ on λ, as defined by

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

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 Rⁿ 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 = H₀¹([−1, +1]; R) and the functional f : X → R given by

$f(u)=\int _{-1}^{+1}u'(x)^{2}\,\mathrm {d} x.$

Without any constraint, the minimum value of f would be 0, attained by u₀(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

$g(u)={\frac {1}{2}}\int _{-1}^{+1}u(x)\,\mathrm {d} x-1.$

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

References

• Zeidler, Eberhard (1995). Applied functional analysis: main principles and their applications. Applied Mathematical Sciences 109. New York, NY: Springer-Verlag. ISBN 0-387-94422-2.

Lagrange multipliers on Banach spaces at PlanetMath.