Constrained Minimization Problems (Method of Lagrange Multipliers)

Abstract

In the calculus of variations we have often to do with the following problem: Given a real valued function f on a nonempty open subset U of a real Banach space E, find the minimum (maximum) of f on all those points x in U which satisfy a certain restriction or constraint. A very important example of such a constraint is that the points have to belong to a level surface of some function g, i.e., have to satisfy g(x) = c where the constant c distinguishes the various level surfaces of the function g. In elementary situations, and typically also in Lagrangian mechanics, one introduces a so-called Lagrange multiplier λ as a new variable and proceeds to minimize the function f(•)+λ(g(•) - c) on the set U. In simple problems (typically finite dimensional) this strategy is successful. The problem is the existence of a Lagrange multiplier.KeywordsBanach SpaceLagrange MultiplierTangent SpaceLevel SurfaceClosed SubspaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.