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.
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\inU\midg(x)=0\inY\}\subseteqU.
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
Df(u0)=λ\circDg(u0). (L)
Since Df(u0) is an element of the dual space X∗, equation (L) can also be written as
Df(u0)=\left(Dg(u0)\right)*(λ),
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(Dg(u0)\right)*(λ)=λ\circDg(u0).
In the case that X and Y are both finite-dimensional (i.e. linearly isomorphic to Rm 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 n = 1, λ is the usual Lagrange multiplier, a real number.
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 and the functional given by
f(u)=
| +1 | |
| \int | |
| -1 |
u'(x)2dx.
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)=
| 1 | |
| 2 |
| +1 | |
| \int | |
| -1 |
u(x)dx-1.
However this problem can be solved as in the finite dimensional case since the Lagrange multiplier
λ
. David Luenberger . Optimization by Vector Space Methods . New York . John Wiley & Sons . 1969 . 0-471-55359-X . Local Theory of Constrained Optimization . 239–270 .