# Lagrange multipliers on Banach spaces

In the field of

calculus of variations inmathematics , 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 ofLagrange 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 space s. Let "U" be an open subset of "X" and let "f" : "U" →**R**be a continuouslydifferentiable 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 "u"

_{0}is a "constrained extremum" of "f", i.e. an extremum of "f" on:$g^\{-1\}\; (0)\; =\; \{\; x\; in\; U\; |\; g(x)\; =\; 0\; in\; Y\; \}\; subseteq\; U.$

Suppose also that the

Fréchet derivative D"f"("u"_{0}) : "X" →**R**of "f" at "u"_{0}is asurjective linear map . Then there exists a**Lagrange multiplier**"λ" : "Y" →**R**in "Y"^{∗}, thedual space to "Y", such that:$mathrm\{D\}\; f\; (u\_\{0\})\; =\; lambda\; circ\; mathrm\{D\}\; g\; (u\_\{0\}).\; quad\; mbox\{(L)\}$

Since D"f"("u"

_{0}) is an element of the dual space "X"^{∗}, equation (L) can also be written as:$mathrm\{D\}\; f\; (u\_\{0\})\; =\; left(\; mathrm\{D\}\; g\; (u\_\{0\})\; ight)^\{*\}\; (lambda),$

where (D"g"("u"

_{0}))^{∗}("λ") is thepullback of "λ" by D"g"("u"_{0}), i.e. the action of theadjoint map (D"g"("u"_{0}))^{∗}on "λ", as defined by:$left(\; mathrm\{D\}\; g\; (u\_\{0\})\; ight)^\{*\}\; (lambda)\; =\; lambda\; circ\; mathrm\{D\}\; g\; (u\_\{0\}).$

**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**R**^{"n"}for somenatural 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" = "H"_{0}^{1}( [−1, +1] ;**R**) and the functional "f" : "X" →**R**given by:$f(u)\; =\; int\_\{-1\}^\{+1\}\; u\text{'}(x)^\{2\}\; ,\; mathrm\{d\}\; x.$

Without any constraint, the minimum value of "f" would be 0, attained by "u"

_{0}("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***

*Wikimedia Foundation.
2010.*

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