In mathematical optimization, the method of Lagrange multipliers (named after Joseph Louis Lagrange) is a strategy for finding the local maxima and minima of a function subject to equality constraints.


(Figure) Find x and y to maximize f(x,y) subject to a constraint (shown in red) g(x,y) = c.


For instance (above), consider the optimization problem

  • maximize f(x,y)
  • subject to g(x,y) = c.

We need both f and g to have continuous first partial derivatives. We introduce a new variable (λ) called a Lagrange multiplier and study the Lagrange function (or Lagrangian) defined by



where the λ term may be either added or subtracted. If f(x0,y0) is a maximum of f(x,y) for the original constrained problem, then there exists λ0 such that (x0,y0,λ0) is a stationary point for the Lagrange function (stationary points are those points where the partial derivatives of Λ are zero, i.e. ∇Λ = 0).



However, not all stationary points yield a solution of the original problem. Thus, the method of Lagrange multipliers yields a necessary condition for optimality in constrained problems. Sufficient conditions for a minimum or maximum also exist.



References


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