Lagrange multipliers

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.

For the case of only one constraint and only two choice variables (as exemplified in Figure 1), consider the optimization problem

We assume that both f and g have continuous first partial derivatives. We introduce a new variable (λ) called a Lagrange multiplier and study the Lagrange function (or Lagrangian or Lagrangian expression) 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 ${\displaystyle {\mathcal {L}}}$ are zero). 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.

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