2
$\begingroup$

Reading Randomized Rounding without Solving the Linear Program paper I came across this definition:

Let $P$ be a convex set in $\mathbb{R}^n$ and let $f$ be a linear function (not necessarily homogeneous) from $P$ to $\mathbb{R}^m$.

Correct me if I am wrong, but $f$ is a non-homogenous function that takes a set of points in $\mathbb{R}^n$ and returns a set of points in $\mathbb{R}^m$?

$\endgroup$
3
$\begingroup$

A homogeneous function $f$ satisfies $f(ax)=af(x)$ for scalars.

When $f(x)=Ax+b$ is linear, this is to say that $b=0$. So a non-homogeneous linear function may have a non-zero constant vector term.

So in your case, $f$ may or may not be homogeneous/non-homogeneous. Strictly speaking it doesn't map $\mathbb{R}^n\to\mathbb{R}^m$, it maps $P\to\mathbb{R}^m$. But since $f$ is linear, the extension is a natural one, and it makes good sense to just think of it as mapping $\mathbb{R}^n\to\mathbb{R}^m$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.