|
|
| Line 1: |
Line 1: |
| 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]]
| |
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.
my web blog ... car warranty (view it)