Lagrange multipliers on Banach spaces

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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.

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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 ∈ U ∣ g ( x ) = 0 ∈ Y } ⊆ U . {\displaystyle 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

D f ( u 0 ) = λ ∘ D g ( u 0 ) . (L) {\displaystyle \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

D f ( u 0 ) = ( D g ( u 0 ) ) ∗ ( λ ) , {\displaystyle \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

( D g ( u 0 ) ) ∗ ( λ ) = λ ∘ D g ( u 0 ) . {\displaystyle \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 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.

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 0 1 ( [ − 1 , + 1 ] ; R ) {\textstyle X=H_{0}^{1}([-1,+1];\mathbb {R} )} and the functional f : X → R {\textstyle f:X\rightarrow \mathbb {R} } given by

f ( u ) = ∫ − 1 + 1 u ′ ( x ) 2 d x . {\displaystyle 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 ) = 1 2 ∫ − 1 + 1 u ( x ) d x − 1. {\displaystyle g(u)={\frac {1}{2}}\int _{-1}^{+1}u(x)\,\mathrm {d} x-1.}

However this problem can be solved as in the finite dimensional case since the Lagrange multiplier λ {\displaystyle \lambda } is only a scalar.

The Lagrange multiplier theorem for Banach spaces

Connection to the finite-dimensional case

Application

See also