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Lagrangian function (where does it come from?)

Here's an explanation to build intuition about Lagrange multipliers. Suppose you need to optimize (maximize or minimize) $f$ under the constraint that $g(x)=0$. Note that if your constraint is of the ...
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Why the Lagrangian function is has different sign depending on the book?

In the case of an equality constraint, it is not important, though one might prefer to write $\mathcal{L}$ and $g$ in such a way that the Lagrange multiplier has a particular meaning. For example, in ...
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Show that Lagrange multiplier method fails to solve $\min(x^2+y^2)$ subject to $(x-1)^3-y^2=0$

First, the method of Lagrange does not apply, it is not that it fails. The gradient of the constraint is zero at the solution. The constraint qualification for the method to apply is that the active ...
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What is the Lagrange Multiplier value?

Yes, indeed. Nice observation. From $\nabla f = \lambda \nabla g$, taking the scalar product with $\nabla g$ follows the equation for $\lambda$ which you wrote.
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Linearizing SOS1

Given that $\lambda \ge 0,$ $$\lambda = 0 \vee l - x = 0$$ can be linearized as $$\lambda \le M z$$ $$L (1-z) \le l - x \le U (1-z)$$ where $z$ is a new binary variable and $M, L, U$ are suitably ...
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Choose the $w_i>0$ such that $\sum_{i=1}^dw_ia_i$ is minimized and $\sum_{i=1}^dw_i=1$

If we find the smallest $a_i$ and set that $w_i$'s weight to 1, that should satisfy the constraint and minimize $\sum_{i=1}^dw_ia_i$. There is no reason to give weight to any $a_i$ than the minimal ...

Complicated Inequality Proof, Variables Subject To Constraint

Too long for comment: Rewrite the inequality as: $$5\sum_{cyc}\frac{a^2bc}{a^2+b^3+c^3}\leq 1,$$ By AM-GM we have $$a^2bc \leq \frac{(1+a)^4}{256},$$ and by the power-means inequality we have  \...
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Finding Extremes of Multivariable Function $f(x,y,z)=xz−yz$
The Lagrange multiplier condition is that to find constrained extrema of $f$ subject to $g_1, g_2 = 0$, we solve the system $\nabla f = \lambda g_1 + \mu g_2$ (or some equivalent condition). In this ...