Lagrange multipliers on Banach spaces
In the field of
calculus of variationsin 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 multipliersas 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 "u"0 is a "constrained extremum" of "f", i.e. an extremum of "f" on
Suppose also that the
Fréchet derivativeD"f"("u"0) : "X" → R of "f" at "u"0 is a surjective linear map. Then there exists a Lagrange multiplier "λ" : "Y" → R in "Y"∗, the dual spaceto "Y", such that
Since D"f"("u"0) is an element of the dual space "X"∗, equation (L) can also be written as
where (D"g"("u"0))∗("λ") is the
pullbackof "λ" by D"g"("u"0), i.e. the action of the adjointmap (D"g"("u"0))∗ on "λ", as defined by
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 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 "m" = "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"01( [−1, +1] ; R) and the functional "f" : "X" → R given by
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
The method of Lagrange multipliers on Banach spaces is required in order to solve this problem.
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