How can one show that $argmin_{a,b}\:\left\{\:\sum \:\left|y-\left(a+bx\right)\right|\right\}$ can lack a unique solution

Question

I'm trying to wrap my head around minimizing the L1 norm, it often possesses infinite solutions despite being a convex function(multiple global optima.) Does that property have a name? and how can one identify the conditions under which a unique solution would prevail?

Answer 1

1. The set of minimizers of convex functions is a convex set. So, if $x$ and $y$ are distinct minimizers of $f$, then every point in the line segment connecting them is also a minimizer (which means that it will have an infinite number of solutions).
2. Uniqueness: If $f$ both (A) has at least one minimizer and is also (B) strictly convex, then the minimizer is guaranteed to be unique. Exercise: show that the L1 norm is not strictly convex.
3. Existence: If $f$ is a convex function which is lower-semicontinuous and also coercive, i.e. $\lim_{\|x\|\to+\infty}f(x)=+\infty$, then a minimizer exists. You can also replace lower-semicontinuity with real-valuedness in that statement (which applies for your function in question).

As far as I know, I don't think there is a name for the set of functions who have multiple minimizers.

A cheeky example demonstrating that the function in your title has multiple minimizers would be $y=a=b=0$. Then everything is a minimizer.