How do you find the points on the parabola $2x={y}^{2}$ that are closest to the point (3,0)?

hotonglamoz
2022-09-20
Answered

How do you find the points on the parabola $2x={y}^{2}$ that are closest to the point (3,0)?

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asked 2022-07-04

What is the Proximal Operator ($\mathrm{Prox}$) of the Pseudo ${L}_{0}$ Norm?

Namely:

${\mathrm{Prox}}_{\lambda {\Vert \cdot \Vert}_{0}}\left(\mathit{y}\right)=\mathrm{arg}\underset{\mathit{x}}{min}\frac{1}{2}{\Vert \mathit{x}-\mathit{y}\Vert}_{2}^{2}+\lambda {\Vert \mathit{x}\Vert}_{0}$

Where ${\Vert \mathit{x}\Vert}_{0}=\mathrm{n}\mathrm{n}\mathrm{z}(x)$, namely teh number of non zeros elements in the vector $\mathit{x}$.

Namely:

${\mathrm{Prox}}_{\lambda {\Vert \cdot \Vert}_{0}}\left(\mathit{y}\right)=\mathrm{arg}\underset{\mathit{x}}{min}\frac{1}{2}{\Vert \mathit{x}-\mathit{y}\Vert}_{2}^{2}+\lambda {\Vert \mathit{x}\Vert}_{0}$

Where ${\Vert \mathit{x}\Vert}_{0}=\mathrm{n}\mathrm{n}\mathrm{z}(x)$, namely teh number of non zeros elements in the vector $\mathit{x}$.

asked 2022-06-16

Let ${x}_{1},...,{x}_{25}>0$ be such that $\sum _{i=1}^{25}{x}_{i}=4350$ and $\sum _{i=1}^{25}{x}_{i}^{2}=757770.25$.

From the first equality alone, we know that at least one of the ${x}_{i}$'s must be less than or equal to $\frac{4350}{25}=174$. From the second equality alone, we know that at least one of the ${x}_{i}$'s must be less than or equal to $\sqrt{\frac{757770.25}{25}}=174.1$, which is less useful than the first bound. My question is whether we can get a better bound, i.e. to find the least upper bound of $min\{{x}_{1},...,{x}_{25}\}$, when we use both equalities together. I appreciate any comments or hints.

From the first equality alone, we know that at least one of the ${x}_{i}$'s must be less than or equal to $\frac{4350}{25}=174$. From the second equality alone, we know that at least one of the ${x}_{i}$'s must be less than or equal to $\sqrt{\frac{757770.25}{25}}=174.1$, which is less useful than the first bound. My question is whether we can get a better bound, i.e. to find the least upper bound of $min\{{x}_{1},...,{x}_{25}\}$, when we use both equalities together. I appreciate any comments or hints.

asked 2022-06-20

I have a doubt regarding a constrained optimisation problem.

Suppose my original constrained minimisation problem is

$\underset{x}{min}f(g(x),x)\phantom{\rule{1em}{0ex}}\text{s.t.}\phantom{\rule{1em}{0ex}}g(x)=3$

I would like to know if this equivalent to solving the unconstrained minimisation problem

$\underset{x}{min}f(3,x)$

If not, when are these two problems equivalent?

Suppose my original constrained minimisation problem is

$\underset{x}{min}f(g(x),x)\phantom{\rule{1em}{0ex}}\text{s.t.}\phantom{\rule{1em}{0ex}}g(x)=3$

I would like to know if this equivalent to solving the unconstrained minimisation problem

$\underset{x}{min}f(3,x)$

If not, when are these two problems equivalent?

asked 2022-08-11

How do you find the dimensions that minimize the amount of cardboard used if a cardboard box without a lid is to have a volume of $8,788{\left(cm\right)}^{3}$?

asked 2022-08-16

What is a tight lower bound to $\sum _{i=1}^{n}\frac{1}{a+{x}_{i}}$ under the restrictions $\sum _{i=1}^{n}{x}_{i}=0$ and $\sum _{i=1}^{n}{x}_{i}^{2}={a}^{2}$ ?

Conjecture: due to the steeper rise of $\frac{1}{a+x}$ for negative $x$, one may keep those values as small as possible. So take $n-1$ values ${x}_{i}=-q$ and ${x}_{n}=(n-1)q$ to compensate for the first condition. The second one then gives ${a}^{2}=\sum _{i=1}^{n}{x}_{i}^{2}={q}^{2}((n-1{)}^{2}+n-1)={q}^{2}n(n-1)$. Hence,

$\sum _{i=1}^{n}\frac{1}{a+{x}_{i}}\ge \frac{n-1}{a(1-1/\sqrt{n(n-1)})}+\frac{1}{a(1+(n-1)/\sqrt{n(n-1)})}$

should be the tight lower bound.

Conjecture: due to the steeper rise of $\frac{1}{a+x}$ for negative $x$, one may keep those values as small as possible. So take $n-1$ values ${x}_{i}=-q$ and ${x}_{n}=(n-1)q$ to compensate for the first condition. The second one then gives ${a}^{2}=\sum _{i=1}^{n}{x}_{i}^{2}={q}^{2}((n-1{)}^{2}+n-1)={q}^{2}n(n-1)$. Hence,

$\sum _{i=1}^{n}\frac{1}{a+{x}_{i}}\ge \frac{n-1}{a(1-1/\sqrt{n(n-1)})}+\frac{1}{a(1+(n-1)/\sqrt{n(n-1)})}$

should be the tight lower bound.

asked 2022-09-23

1- What is Optimization? How many methods are there to calculate it?

2- What do we mean by an objective function? What do we mean by constraints?

3- Give three practical examples (physical or engineering) of a target function with a constraint

2- What do we mean by an objective function? What do we mean by constraints?

3- Give three practical examples (physical or engineering) of a target function with a constraint

asked 2022-05-09

Could you help me solve this problem please ?

1. Maximize ${x}^{t}y$ with constraint ${x}^{t}Qx\le 1$ (where $Q$ is definite positive)

What I tried : I tried using KKT but I don't know why I get $-\sqrt{{y}^{t}{Q}^{-1}y}$ as the maximum instead of $\sqrt{{y}^{t}{Q}^{-1}y}$ (which I believe is the maximum). Also, since ${x}^{t}y$ is linear (convex and concave), I don't know how to conclude...

2. Conclude that $({x}^{t}y{)}^{2}\le ({x}^{t}Qx)({y}^{t}{Q}^{-1}y)$$\mathrm{\forall}x,y$ (generalized CS)

1. Maximize ${x}^{t}y$ with constraint ${x}^{t}Qx\le 1$ (where $Q$ is definite positive)

What I tried : I tried using KKT but I don't know why I get $-\sqrt{{y}^{t}{Q}^{-1}y}$ as the maximum instead of $\sqrt{{y}^{t}{Q}^{-1}y}$ (which I believe is the maximum). Also, since ${x}^{t}y$ is linear (convex and concave), I don't know how to conclude...

2. Conclude that $({x}^{t}y{)}^{2}\le ({x}^{t}Qx)({y}^{t}{Q}^{-1}y)$$\mathrm{\forall}x,y$ (generalized CS)