New answers tagged lagrange-multiplier
3
votes
Accepted
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 ...
- 188k
1
vote
Lagrange multiplier for QCQP with $1$ equality constraint
($\lambda = 1$, $x=0$ and $y=\pm 1$) Or ($\lambda = -2$, $y=0$ and $x=\pm \sqrt 2$)
- 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:
$$ \inf_A \;\bigg\{\sum_{i=1}^n (y - b - ...
- 43
3
votes
How to find the maximum area of a quadrilateral, when three of the sides add up to 24?
Reflect the quadrilateral across the fourth side to produce a hexagon with perimeter $48$. Of all hexagons with a given perimeter, the one with the greatest area is the regular hexagon (proof). So the ...
- 2,769
0
votes
How to find the maximum area of a quadrilateral, when three of the sides add up to 24?
Denote the four sides of the quadrilateral by $w = DA$, $x = AB$, $y = BC$, $z = CD$.
Let $d = AC$ be the length of one diagonal. This splits the quadrilateral into two triangles, $\triangle ABC$ and ...
- 9,658
0
votes
Mathematical explanation of the intuition behind the Lagrange multiplier
From the f.o.c. for utility maximization under a budget constraint, we have
$$\frac{\partial U}{\partial x} = \lambda p_x,\;\;\;\frac{\partial U}{\partial y} = \lambda p_y$$
that holds at the optimal ...
- 9,960
2
votes
pseudo C-S inequality?
Hint: You can also use Hölder's Inequality, like so:
$$\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$$
- 42.9k
4
votes
pseudo C-S inequality?
Using Lagrange multiplier method:
$$\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\...
- 33.4k
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[...
- 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:
$$
\sqrt[4]{\...
- 70.7k
0
votes
pseudo C-S inequality?
Define:
$$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}$$
Now you can use C-S ...
- 5,886
0
votes
$λ\geq 0$ constraint in Lagrangian
An inequality constrained problem
\begin{equation*}
\begin{aligned}
& \underset{x}{\text{minimize}}
& & f(x) \\
& \text{subject to}
& & g_i(x) \leq 0, \; i = 1, \ldots, m.
\end{...
- 71
2
votes
Accepted
Why must Lagrange multipliers for inequality constraints be non-negative?
An inequality constrained problem
\begin{equation*}
\begin{aligned}
& \underset{x}{\text{minimize}}
& & f(x) \\
& \text{subject to}
& & g_i(x) \leq 0, \; i = 1, \ldots, m.
\end{...
- 71
0
votes
$λ\geq 0$ constraint in Lagrangian
I will try to give a geometric idea about the convention $\lambda \ge 0$
Suppose we have the problem
$$
\max f(x)\ \ \text{s. t.}\ \ g_k(x) \le 0
$$
In this case to qualify a stationary point $x^*$ we ...
- 26.5k
Top 50 recent answers are included