then how we says that this is a nls where the one property of norm is not satisfied that norm(f)=0 then f=0 a.e. but actually not zero then how it says that this is a nls??

# Tag: norm

## convex optimization – How do you call a linear programming problem when the solution should be “constrained” to a norm?

(apologies for the n00b question)

Let’s say we have a vector of length $n$, with to-be-determined values: $a_1, a_2, …,a_n$.

And we have information that partial sums of these elements are equal to something, say:

$$

a_1 + a_2 + … + a_{k_1} = A_{1} \

a_{k_1+1} + a_{k_1+2} + … + a_n = B_1 \

a_1 + a_2 + … + a_{k_2} = A_2 \

a_{k_2+1} + a_{k_2+2} + … + a_n = B_2 \

…\

a_1 + a_2 + … + a_{k_m} = A_m \

a_{k_m+1} + a_{k_m+2} + … + a_n = B_m \

$$

Where $m<<n$: so we have much fewer such $m$ equations/constraints than the $n$ unknown values $a_i$.

If we want to know which combination of $a_i$ values can solve these equations, there are probably infinite many such combinations (or 0). So I’d like to add two constraints to this:

- $a_i>0$ for any i.
- I want the solution with $a_i$ values that are as “similar” to each other as possible. For example, keeping $sum_{i=1}^n (a_i – bar a)^2$ as small as possible (L2 norm). Where $bar a = sum frac{a_i}{n}$.

How is such optimization problem called? (would also love to know how to solve it, but I assume that once I have the name, I can find solvers).

## memory hardware – Assembly Language Step By Step: Why deviate from the norm and design computers where the presence of voltage encodes a 0 bit?

That the presence of voltage across a switch encodes 1_{2} is purely arbitrary… Jeff Duntemann’s book mentions:

We could as well have said that the lack of voltage indicates a binary

1 and vice versa (andcomputers have been built this wayfor obscure

reasons)

I find this (italicized part) quite fascinating. It would be great if someone could shed some light on what “obscure” reason(s) may motivate people to do so?

## How is a norm base "A" defined?

Assuming A is a matrix and v a vector.

What’s the name of the following norm and how is it defined?

The norm I am talking about

## fa.functional analysis – Determine the norm of a continuous linear operator $T:L^1[a,b]to L^1[a,b]$

I have encountered the following exercise:

Let $K(x,y)$ be a measurable function on $(a,b)times (a,b)$. The

function $I:yin (a,b) mapsto int_a^b |K(x,y)|, text{d}x in (0,+infty)$ belongs to $L^infty (a,b)$. Define an operator $T$ by

begin{equation} Tf: xin (a,b)mapstoint_a^b K(x,y) f(y) , text{d} y in (-infty,+infty),

qquad forall fin L^1(a,b).

end{equation} Show that $T$ is a

continuous linear operator from $L^1(a,b)$ to $L^1(a,b)$, and

begin{equation} |T| = |I|_infty. end{equation}

Clearly, by Fubini’s theorem, we obtain

begin{align*}

|Tf|_1& = int_a^b left| int_a^b K(x,y) f(y) , text{d} yright|, text{d}x

leq int_a^b int_a^b left| K(x,y) f(y)right|, text{d} y , text{d}x

\ &=int_a^b |f(y)|int_a^b left| K(x,y)right| , text{d}x , text{d} y

= int_a^b |f(y) I(y)|, text{d} y \& leq |I|_infty |f|_1

end{align*}

for all $fin L^1(a,b)$, thus $|T|leq |I|_infty$.

But I have no idea how to prove the converse inequality. I know we just need to find some $f_varepsilonin L^1(a,b)$ such that $|Tf_varepsilon|_1 geq (|I|_infty-varepsilon) |f_varepsilon|_1$ for any $varepsilon>0$. But I have difficluty in finding such an appropriate function arbitrarily close to attaining the infinity norm of $I$.

Any ideas would be greatly appreciated.

## linear algebra – Follow-Up to “Least Squares with Euclidean $(L_2)$ Norm Constaint”

I would have a couple of follow-up questions to the following answer made by Royi:

The setup is the following:

$$ begin{alignat*}{3} text{minimize} & quad & frac{1}{2} left| A

x – b right|_{2}^{2} \ text{subject to} & quad & {x}^{T} x leq 1

end{alignat*} $$The Lagrangian is given by:

$$ L left( x, lambda right) = frac{1}{2} left| A x – b

right|_{2}^{2} + frac{lambda}{2} left( {x}^{T} x – 1 right) $$The KKT Conditions are given by:

$$ begin{align*} nabla L left( x, lambda right) = {A}^{T} left(

A x – b right) + lambda x & = 0 && text{(1) Stationary Point} \

lambda left( {x}^{T} x – 1 right) & = 0 && text{(2) Slackness} \

{x}^{T} x & leq 1 && text{(3) Primal Feasibility} \ lambda & geq

0 && text{(4) Dual Feasibility} end{align*} $$

1.) Why is the “Slackness” condition fulfilled, i.e. $lambdaleft( x^{T}x – 1 right) = 0$? After all, we only know that $leftvertleftvert xrightvertrightvert_{2}^{2} leq 1$, correct?

2.) Very related: Where do we know from that $lambdageq 0$?

3.) Here, the Lagrangian was defined by $mathcal L left( x, lambda right) := frac{1}{2} left| A x – b right|_{2}^{2} + frac{lambda}{2} left( {x}^{T} x – 1 right)$. One can also define the Lagrangian by $mathcal L left( x, lambda right) := frac{1}{2} left| A x – b right|_{2}^{2} – frac{lambda}{2} left( {x}^{T} x – 1 right)$, it shouldn’t matter at the end. But in the latter case, would $lambdageq 0$ still hold?

4.) I would have a very general question concerning the KKT theorem: Can one also apply them when one deals with equality constraints?

## Scalar-by-vector derivative involving L2 norm and Hadamard product

I have a function $f(mathbf{x})$: $mathbb{R}^N rightarrow mathbb{R}$ given by:

$f(mathbf{x}) = lvert mathbf{A}(mathbf{x} circ mathbf{x})-mathbf{b} rvert^2$.

with $mathbf{A} in mathbb{R}^{Ntimes N}$ a constant matrix, $mathbf{b}$ the known vector $in mathbb{R}^N$, $lvert cdot rvert^2$ the $ell_2$ norm and $circ$ the hadamard product. I would like to know the expression about the gradient of $f$ with respect to $mathbf{x}$.

## L1 Norm Optimization Solution – Mathematics Stack Exchange

I am trying to find the solution for the following optimization problem:

$max_{w} {z^Tw – lambda ||w – w_0||_1}$

where $z, w, w_0 in R^{Nx1}$ and $z, w_0$ are known.

We let $s$ be the subgradient of $lambda ||w – w_0||_1, hspace{2mm} s in { {u | u^T(w – w_0) = lambda ||w – w_0||_1, ||u||_{infty} leq lambda } }$.

If we write $lambda ||w – w_0||_1 = max_{||s||_{infty} leq lambda} s^T(w – w_0)$, the objective function can now be written as:

$max_{w} min_{||s||_{infty} leq lambda} {z^Tw – s^T(w – w_0)}$

$min_{||s||_{infty} leq lambda} max_{w} {z^Tw – s^T(w – w_0)}$.

The first order condition for the inner objective function is:

$0 = z – s hspace{1cm} (1)$

Substituting (1) into the objective function,

$min_{||s||_{infty} leq lambda} s^Tw_0$.

Hence, this objective function can be simplified as,

$min_s s^Tw_0 text{ s.t } -lambda leq s leq lambda$.

However, if all the steps above are correct (which I am not very certain about), I am not sure how the solution of *s* from the “simplified” optimization above will help me solve for *w*.

The sequence of steps taken to solve the initial optimization problem was inspired by the following paper.

Any help on this is very much appreciated. Thank you.

## mg.metric geometry – A generalized norm function in $mathbb{R}^n$

We defined a new norm. The norm of $x in mathbb{R}^n$ is defined as

$$ N_P(x) = min {t geq 0 : x in tcdot P} enspace,$$

where $P$ is a centrally symmetric and convex body centered at the origin point.

We prove that it is a norm.

1.Identity of indiscernibles.

Obviously, $N_P(x)=0 Leftrightarrow x=0$.

2.Absolutely scalable.

Because of centrally symmetric property, $N_P(ax)=|a|N_P(x)$.

3.Triangle inequality.

Denote $N_P(x+y), N_P(x),N_P(y)$ as $t_0,t_1,t_2$ respectively. And let vectors $x+y,x,y$ go from the origin point and hit the border of $P$ at $a,b,c$ respectively.

Therefore $(x+y)=x+y$ implies $t_0vec{a}=t_1vec{b}+t_2vec{c}$, implies $vec{a}=frac{t_1}{t_0}vec{b}+frac{t_2}{t_0}vec{c}$

Suppose $t_0 > t_1+t_2$, thus $0leq frac{t_1}{t_0}+frac{t_2}{t_0}<1$.

However, this contradict to the convex property because border $bac$ is not convex. QED

We realized this new norm consists all possible norms in $mathbb{R}^n$, including $ell_p$.

Because for any norm $N(cdot)$, define $P={ x : N(x)leq 1}$, one can verify that $N_P=N$. It shows a simple fact: a norm is equivalent to the space which has unit norm.

Our question is, did anyone discover it before? What is the name? We googled it but did not get answers.

## linear algebra – Derivative of a norm

I learned not use the Norm() function when computing vector derivative, so I use the dot product instead:

```
In: D(x.x, x)
Out: 1.x + x.1
```

What does the result mean? Is 1=(1, 1, .., 1) here? Why can’t it show just `2x`

as the result?

And Mathematica won’t resolve it when I define `x`

?

```
In: 1.x + x .1 /. x -> {3, 4}
Out: {0.3 + 1.{3, 4}, 0.4 + 1.{3, 4}}
```