information:extraction, information:filtering
Interpretation of the Lagrange multipliers

In this section, we modify the constraint equations from the form g i (x) = 0 {\displaystyle g_{i}({\bf {x}})=0} to the form g i (x) = c i , {\displaystyle \ g_{i}({\bf {x}})=c_{i}\,} where the c i {\displaystyle \ c_{i}\ } are m real constants that are considered to be additional arguments of the Lagrangian expression L {\displaystyle {\mathcal {L}}}.

Often the Lagrange multipliers have an interpretation as some quantity of interest. For example, by parametrising the constraint's contour line, that is, if the Lagrangian expression is L (x 1, x 2, …; λ 1, λ 2, …; c 1, c 2, …) = f (x 1, x 2, …) + λ 1 (c 1 − g 1 (x 1, x 2, …)) + λ 2 (c 2 − g 2 (x 1, x 2, …)) + ⋯ {\displaystyle {\begin{aligned}&{\mathcal {L}}(x_{1},x_{2},\ldots ;\lambda _{1},\lambda _{2},\ldots;c_{1},c_{2},\ldots)\\={}&f(x_{1},x_{2},\ldots)+\lambda _{1}(c_{1}-g_{1}(x_{1},x_{2},\ldots))+\lambda _{2}(c_{2}-g_{2}(x_{1},x_{2},\dots))+\cdots \end{aligned}}} then ∂ L ∂ c k = λ k. {\displaystyle \ {\frac {\partial {\mathcal {L}}}{\partial c_{k}}}=\lambda _{k}~.}

So, λ is the rate of change of the quantity being optimized as a function of the constraint parameter.As examples, in Lagrangian mechanics the equations of motion are derived by finding stationary points of the action, the time integral of the difference between kinetic and potential energy. Thus, the force on a particle due to a scalar potential, F = −∇ V, can be interpreted as a Lagrange multiplier determining the change in action (transfer of potential to kinetic energy) following a variation in the particle's constrained trajectory. In control theory this is formulated instead as costate equations.

Moreover, by the envelope theorem the optimal value of a Lagrange multiplier has an interpretation as the marginal effect of the corresponding constraint constant upon the optimal attainable value of the original objective function: If we denote values at the optimum with a star (⋆ {\displaystyle \star }), then it can be shown that d f (x 1 ⋆ (c 1, c 2, …), x 2 ⋆ (c 1, c 2, …), …) d c k = λ ⋆ k. {\displaystyle {\frac {\ \operatorname {d} f\left(\ x_{1\star }(c_{1},c_{2},\dots),\ x_{2\star }(c_{1},c_{2},\dots),\ \dots \ \right)\ }{\operatorname {d} c_{k}}}=\lambda _{\star k}~.}

For example, in economics the optimal profit to a player is calculated subject to a constrained space of actions, where a Lagrange multiplier is the change in the optimal value of the objective function (profit) due to the relaxation of a given constraint (e.g. through a change in income); in such a context λ ⋆ k {\displaystyle \ \lambda _{\star k}\ } is the marginal cost of the constraint, and is referred to as the shadow price.