Lagrange multipliers on 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

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

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

:$\backslash mathrm\{D\}\; f\; (u\_\{0\})\; =\; \backslash lambda\; \backslash circ\; \backslash mathrm\{D\}\; g\; (u\_\{0\}).\; \backslash quad\; \backslash mbox\{(L)\}$

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

:$\backslash mathrm\{D\}\; f\; (u\_\{0\})\; =\; \backslash left(\; \backslash mathrm\{D\}\; g\; (u\_\{0\})\; \backslash right)^\{*\}\; (\backslash lambda),$

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

:$\backslash left(\; \backslash mathrm\{D\}\; g\; (u\_\{0\})\; \backslash right)^\{*\}\; (\backslash lambda)\; =\; \backslash lambda\; \backslash circ\; \backslash mathrm\{D\}\; g\; (u\_\{0\}).$

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 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 $X\; =\; H\_0^1([-1,+1];\backslash mathbb\{R\})$ and the functional $f\; :\; X\; \backslash rightarrow\; \backslash mathbb\{R\}$ given by

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

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

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

However this problem can be solved as in the finite dimensional case since the Lagrange multiplier $\backslash lambda$ is only a scalar.

• Pontryagin's minimum principle, Hamiltonian method in calculus of variations

• {{cite book |first=David G. |last=Luenberger |authorlink=David Luenberger |title=Optimization by Vector Space Methods |location=New York |publisher=John Wiley & Sons |year=1969 |isbn=0-471-55359-X |chapter=Local Theory of Constrained Optimization |pages=239–270 }}
• {{cite book | title=Applied functional analysis: Variational Methods and Optimization | url=https://link.springer.com/book/10.1007/978-1-4612-0821-1 | last=Zeidler | first=Eberhard | publisher=Springer-Verlag | year=1995 | isbn=978-1-4612-0821-1 | series=Applied Mathematical Sciences 109 | volume=109 | location=New York, NY | doi=10.1007/978-1-4612-0821-1 }} (See Section 4.14, pp.270–271.)
• {{cite book | title=Nonlinear Functional Analysis and its Applications III| url= | last=Zeidler | first=Eberhard | publisher=Springer-Verlag | year=1984 | isbn=978-1-4612-6971-7 | location=New York, NY | chapter = Lagrange Multipliers and Eigenvalue Problems | pages= 273-300}}

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Category:Calculus of variations