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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?2010-09-25
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    @figuringout Sounds like what you're looking for is the envelope theorem.2014-05-27

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