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
- [math]\displaystyle{ g^{-1} (0) = \{ x \in U \mid g(x) = 0 \in Y \} \subseteq U. }[/math]
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
- [math]\displaystyle{ \mathrm{D} f (u_{0}) = \lambda \circ \mathrm{D} g (u_{0}). \quad \mbox{(L)} }[/math]
Since Df(u0) is an element of the dual space X∗, equation (L) can also be written as
- [math]\displaystyle{ \mathrm{D} f (u_{0}) = \left( \mathrm{D} g (u_{0}) \right)^{*} (\lambda), }[/math]
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
- [math]\displaystyle{ \left( \mathrm{D} g (u_{0}) \right)^{*} (\lambda) = \lambda \circ \mathrm{D} g (u_{0}). }[/math]
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 [math]\displaystyle{ X = H_0^1([-1,+1];\mathbb{R}) }[/math] and the functional [math]\displaystyle{ f : X \rightarrow \mathbb{R} }[/math] given by
- [math]\displaystyle{ f(u) = \int_{-1}^{+1} u'(x)^{2} \, \mathrm{d} x. }[/math]
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
- [math]\displaystyle{ g(u) = \frac{1}{2} \int_{-1}^{+1} u(x) \, \mathrm{d} x - 1. }[/math]
However this problem can be solved as in the finite dimensional case since the Lagrange multiplier [math]\displaystyle{ \lambda }[/math] is only a scalar.
See also
- Pontryagin's minimum principle, Hamiltonian method in calculus of variations
References
- Luenberger, David G. (1969). "Local Theory of Constrained Optimization". Optimization by Vector Space Methods. New York: John Wiley & Sons. pp. 239–270. ISBN 0-471-55359-X.
- Zeidler, Eberhard (1995). Applied functional analysis: Variational Methods and Optimization. Applied Mathematical Sciences 109. 109. New York, NY: Springer-Verlag. doi:10.1007/978-1-4612-0821-1. ISBN 978-1-4612-0821-1. https://link.springer.com/book/10.1007/978-1-4612-0821-1. (See Section 4.14, pp.270–271.)
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