en>K9re11: added Category:Properties of groups using HotCat

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added Category:Properties of groups using HotCat

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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 [[constraint (mathematics)\|constrained]] [[optimization (mathematics)\|optimization problems]]. The ~~method~~ is ~~a generalization of the classical method of [[Lagrange multipliers]] as used~~ to ~~find [[extremum\|extrema]] of a [[function (mathematics)\|function]] of finitely many variables.~~		Friends contact him Royal Cummins. The factor she adores most is to perform handball but she can't make it her occupation. Delaware is the only location I've been residing in. I am a manufacturing and distribution officer.<br><br>my web blog ... car warranty ([http://www.Myccos.com/UserProfile/tabid/61/userId/430/Default.aspx view it])

	~~==The Lagrange multiplier theorem for Banach spaces==~~
	~~Let~~ '~~'X'' and ''Y'' be [[real number\|real]] [[Banach space]]s~~. Let ''U'' be an [[open set\|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 ''u''<sub>0</sub> is a ''constrained extremum'' of ''f'', i.e. an extremum of ''f'' on~~

	~~:<math>g^{-1} (0) = \{ x \~~in ~~U \mid g(x) = 0 \in Y \} \subseteq U~~.~~</math>~~

	~~Suppose also that the [[Fréchet derivative]] D''g''(''u''<sub>0</sub>) : ''X'' → ''Y'' of ''g'' at ''u''<sub>0</sub> is~~ a ~~[[surjective]] [[linear map]]. Then there exists a '''Lagrange multiplier''' ''λ'' : ''Y'' → '''R''' in ''Y''<sup>∗</sup>, the [[dual space]] to ''Y'', such that~~

	~~:<math>\mathrm{D} f (u_{0}) = \lambda \circ \mathrm{D} g (u_{0})~~. ~~\quad \mbox{(L)}</math>~~

	~~Since D''f''(''u''<sub>0</sub>) is an element of the dual space ''X''<sup>∗</sup>, equation (L) can also be written as~~

	~~:<math>\mathrm{D} f (u_{0}) = \left( \mathrm{D} g (u_{0}) \right)^{*} (\lambda),</math>~~

	~~where (D''g''(''u''<sub>0</sub>))<sup>∗</sup>(''λ'') is the [[pullback]] of ''λ'' by D''g''(''u''~~<~~sub~~>0<~~/sub~~>~~), i~~.e. ~~the action of the [[adjoint]]{{dn\|date=December 2013}} map (D''g''(''u''<sub>0</sub>))<sup>∗</sup> on ''λ'', as defined by~~

	~~:<math>\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.~~ [~~[linear isomorphism\|linearly isomorphic]] to '''R'''<sup>''m''<~~/~~sup> and '''R'''<sup>''n''<~~/sup> for some [[natural numbers]] ''m'' and ''n'') then writing out equation (L) in [[matrix (mathematics)\|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''<sub>0<~~/~~sub><sup>1<~~/~~sup>([−1, +1]; '''R''') and the functional ''f'' : ''X'' → '''R''' given by~~

	~~:<math>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 ''u''<sub>0<~~/sub>(''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>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> \lambda <~~/~~math> is only a scalar~~.

	~~==See also==~~
	* [[Pontryagin's minimum principle]], Hamiltonian method in calculus of variations

	~~==References==~~
	* {{cite book \| last=Zeidler \| first=Eberhard \| title=Applied functional analysis: main principles and their applications \| series=Applied Mathematical Sciences 109 \| publisher=Springer-Verlag \| location=New York, NY \| year=1995 \| isbn=0-387-94422-2 }}

	~~{{PlanetMath attribution\|id=7329\|title=Lagrange multipliers on Banach spaces}}~~

	~~[[Category:Calculus of variations]]~~
	~~[[Category:Mathematical optimization]~~]

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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 [[constraint (mathematics)|constrained]] [[optimization (mathematics)|optimization problems]]. The method is a generalization of the classical method of [[Lagrange multipliers]] as used to find [[extremum|extrema]] of a [[function (mathematics)|function]] of finitely many variables.

==The Lagrange multiplier theorem for Banach spaces==
Let ''X'' and ''Y'' be [[real number|real]] [[Banach space]]s. Let ''U'' be an [[open set|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 ''u''0 is a ''constrained extremum'' of ''f'', i.e. an extremum of ''f'' on

:<math>g^{-1} (0) = \{ x \in U \mid g(x) = 0 \in Y \} \subseteq U.</math>

Suppose also that the [[Fréchet derivative]] D''g''(''u''0) : ''X'' → ''Y'' of ''g'' at ''u''0 is a [[surjective]] [[linear map]]. Then there exists a '''Lagrange multiplier''' ''λ'' : ''Y'' → '''R''' in ''Y''∗, the [[dual space]] to ''Y'', such that

:<math>\mathrm{D} f (u_{0}) = \lambda \circ \mathrm{D} g (u_{0}). \quad \mbox{(L)}</math>

Since D''f''(''u''0) is an element of the dual space ''X''∗, equation (L) can also be written as

:<math>\mathrm{D} f (u_{0}) = \left( \mathrm{D} g (u_{0}) \right)^{*} (\lambda),</math>

where (D''g''(''u''0))∗(''λ'') is the [[pullback]] of ''λ'' by D''g''(''u''0), i.e. the action of the [[adjoint]]{{dn|date=December 2013}} map (D''g''(''u''0))∗ on ''λ'', as defined by

:<math>\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. [[linear isomorphism|linearly isomorphic]] to '''R'''''m'' and '''R'''''n'' for some [[natural numbers]] ''m'' and ''n'') then writing out equation (L) in [[matrix (mathematics)|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''01([−1, +1]; '''R''') and the functional ''f'' : ''X'' → '''R''' given by

:<math>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 ''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

:<math>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> \lambda </math> is only a scalar.

==See also==
* [[Pontryagin's minimum principle]], Hamiltonian method in calculus of variations

==References==
* {{cite book | last=Zeidler | first=Eberhard | title=Applied functional analysis: main principles and their applications | series=Applied Mathematical Sciences 109 | publisher=Springer-Verlag | location=New York, NY | year=1995 | isbn=0-387-94422-2 }}

{{PlanetMath attribution|id=7329|title=Lagrange multipliers on Banach spaces}}

[[Category:Calculus of variations]]
[[Category:Mathematical optimization]]

Free-by-cyclic group - Revision history

en>K9re11: added Category:Properties of groups using HotCat

en>Qetuth: more specific stub type