Lagrange Multipliers
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Here's part of the explanation of constrained optimization from Hughes-Hallett et al. *Applied Calculus* 4/e (p. 383):
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#### QUESTION
Why does this method work?
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This is an introduction to an algorithm for solving constrained optimization problems.
Before introducing an algorithm, better to introduce some context and concepts:
* Objective and constraint functions. Why do we distinguish?
* The constraint has an impact (sometimes) on the output of the objective function. What's the relationship? (Shadow prices)
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Here's another explanation from the same book (p. 382).
This makes it clearer why there is a tangency relationship at the optimum. But it still treats the constraint as an equation rather than as a function.