Editing Lagrange multiplier (section)

==Interpretation of the Lagrange multipliers==
In this section, we modify the constraint equations from the form <math>g_i({\bf x}) = 0</math> to the form <math>\ g_i({\bf x}) = c_i\ ,</math> where the <math>\ c_i\ </math> are {{mvar|m}} real constants that are considered to be additional arguments of the Lagrangian expression <math>\mathcal{L}</math>.

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
<math display="block">
\begin{align}
& \mathcal{L}(x_1, x_2, \ldots;\lambda_1, \lambda_2, \ldots; c_1, c_2, \ldots) \\[4pt]
= {} & 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{align}
</math>
then
<math display="block">\ \frac{\partial \mathcal{L}}{\partial c_k} = \lambda_k ~.</math>

So, {{math|''λ<sub>k</sub>''}} 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 (physics)|action]], the time integral of the difference between kinetic and potential energy.  Thus, the force on a particle due to a scalar potential, {{math|''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 (<math>\star</math>), then it can be shown that
<math display="block"> \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} ~.</math>

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 <math>\ \lambda_{\star k}\ </math> is the [[marginal cost]] of the constraint, and is referred to as the [[shadow price]].<ref>{{cite book |first=Avinash K. |last=Dixit |author-link=Avinash Dixit |chapter=Shadow Prices |title=Optimization in Economic Theory |location=New York |publisher=Oxford University Press |edition=2nd |year=1990 |pages=40–54 |isbn=0-19-877210-6 |chapter-url=https://books.google.com/books?id=dHrsHz0VocUC&pg=PA40 }}</ref>