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
Famous quotes containing the words general, envelope and/or theorem:
“Anti-Nebraska, Know-Nothings, and general disgust with the powers that be, have carried this county [Hamilton County, Ohio] by between seven and eight thousand majority! How people do hate Catholics, and what a happiness it was to show it in what seemed a lawful and patriotic manner.”
—Rutherford Birchard Hayes (1822–1893)
“Geroge Peatty: I’m gonna have it, Sherry. Hundreds of thousands, maybe a half million.
Sherry Peatty: Of course you are, darling. Did you put the right address on the envelope when you sent it to the North Pole?”
—Stanley Kubrick (b. 1928)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)