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In constrained Lagrangian optimization what is a general way to figure out how the optimal point varies with respect to parameters of the constraints? For example, maximize $x\cdot y$ when $x + y \lt k$, and I wanted to find $\large\frac{dx^{*}}{dk}$ and $\large\frac{dy^{*}}{dk}$ where $(x^{*},y^{*})$ is the optimal point. How would I do that?

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The example problem you have given has no solution since you can take x and y to be arbitrarily large negative numbers. Or do you also have a non-negativity constraint? – Jyotirmoy Bhattacharya Sep 25 '10 at 5:53
@figuringout Sounds like what you're looking for is the envelope theorem. – ben May 27 '14 at 4:05

The optimal point may not be differentiable at a value of $k$ where a constraint goes from binding to non-binding. At any other value of $k$ just differentiate with respect to $k$ the (equality) first order conditions for the constraints which bind and solve for $\partial x^*/\partial k$ etc.

Take a look at any standard optimization book, for eg. Luenberger Linear and Non-linear Programming or Sundaram's A First Course in Optimization Theory,

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I would just add that even in the case where a constraint goes from binding to nonbinding that the optimal point ought to have both one-sided derivatives with respect to $k$ existing. They just won't be equal to each other (and so the derivative itself won't exist). – Mike Spivey Oct 19 '10 at 19:47

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