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3 votes
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Lagrange multipliers and not tangent contour lines

Assuming the function indicated here waves nicely and sine-like in the way hinted at in the drawing (say we have $f(x,y)=\cos(x+y)+3$), the Lagrange multiplier method will find all the red dots in the ...
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1 vote

Lagrange multiplier for QCQP with $1$ equality constraint

($\lambda = 1$, $x=0$ and $y=\pm 1$) Or ($\lambda = -2$, $y=0$ and $x=\pm \sqrt 2$)
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  • 5,902
0 votes

Constrained Optimization and Integer Programming

I am going to assume your problem is to solve the sum of the objective you mention as noted in the comments minimizing a vector is not necessarily well posed: $$ \inf_A \;\bigg\{\sum_{i=1}^n (y - b - ...
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  • 43
3 votes

How to find the maximum area of a quadrilateral, when three of the sides add up to 24?

Reflect the quadrilateral across the fourth side to produce a hexagon with perimeter $48$. Of all hexagons with a given perimeter, the one with the greatest area is the regular hexagon (proof). So the ...
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0 votes

How to find the maximum area of a quadrilateral, when three of the sides add up to 24?

Denote the four sides of the quadrilateral by $w = DA$, $x = AB$, $y = BC$, $z = CD$. Let $d = AC$ be the length of one diagonal. This splits the quadrilateral into two triangles, $\triangle ABC$ and ...
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0 votes

Mathematical explanation of the intuition behind the Lagrange multiplier

From the f.o.c. for utility maximization under a budget constraint, we have $$\frac{\partial U}{\partial x} = \lambda p_x,\;\;\;\frac{\partial U}{\partial y} = \lambda p_y$$ that holds at the optimal ...
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2 votes

pseudo C-S inequality?

Hint: You can also use Hölder's Inequality, like so: $$\left(8x^4+27y^4+64z^4 \right) \cdot \left(\frac12+\frac13+\frac14 \right)^3 \geqslant \left(|x|+|y|+|z|\right)^4 \geqslant (x+y+z)^4$$
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4 votes

pseudo C-S inequality?

Using Lagrange multiplier method: $$\frac{\partial}{\partial x}\left(8x^4+27y^4+64z^4-\lambda(x+y+z-\frac{13}4)\right)=0\\4\cdot8x^3=\lambda$$ Similarly for $y$ and $z$ you get: $$4\cdot 27y^3=\lambda\...
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3 votes

pseudo C-S inequality?

Apply AM-GM inequality repeatedly three times: $8x^4+\dfrac{243}{2}=8x^4 + \dfrac{81}{2}+\dfrac{81}{2}+\dfrac{81}{2} \ge 4\sqrt[4]{8x^4\cdot \dfrac{81}{2}\cdot \dfrac{81}{2}\cdot \dfrac{81}{2}}=4\sqrt[...
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  • 3,875
4 votes

pseudo C-S inequality?

Let $\,u = 2x, v=3y, w=4z\,$ then the problem is to minimize $u^4 / 2 + v^4 / 3 + w^4 / 4$ under the constraint $u / 2 + v / 3 + w / 4 = 13/4$. By the weighted power means inequality: $$ \sqrt[4]{\...
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0 votes

pseudo C-S inequality?

Define: $$u^4=8x^4,~v^4=27y^4,~w=64z^4$$ $$f=u^4+v^4+w^4$$ Constraint becomes: $$8^{-1/4}u+27^{-1/4}v+64^{-1/4}w=13/4$$ So we have: $$a=8^{-1/4},~~b=27^{-1/4},~~c=64^{-1/4}$$ Now you can use C-S ...
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  • 5,886
0 votes

$λ\geq 0$ constraint in Lagrangian

An inequality constrained problem \begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} & & f(x) \\ & \text{subject to} & & g_i(x) \leq 0, \; i = 1, \ldots, m. \end{...
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  • 71
2 votes
Accepted

Why must Lagrange multipliers for inequality constraints be non-negative?

An inequality constrained problem \begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} & & f(x) \\ & \text{subject to} & & g_i(x) \leq 0, \; i = 1, \ldots, m. \end{...
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  • 71
0 votes

$λ\geq 0$ constraint in Lagrangian

I will try to give a geometric idea about the convention $\lambda \ge 0$ Suppose we have the problem $$ \max f(x)\ \ \text{s. t.}\ \ g_k(x) \le 0 $$ In this case to qualify a stationary point $x^*$ we ...
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