There also exists a version of the theorem, called the general envelope theorem, used in constrained optimization problems which relates the partial derivatives of the optimal-value function to the partial derivatives of the Lagrangian function.
We are considering the following optimization problem in formulating the theorem (max may be replaced by min, and all results still hold):
Which gives the Lagrangian function:
Where:
- is the dot product
Then the general envelope theorem is:
Note that the Lagrange multipliers are treated as constants during differentiation of the Lagrangian function, then their values as functions of the parameters are substituted in afterwards.
Read more about this topic: Envelope Theorem
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