The following graph represents the sublevel set $ S_{-1} = \{(x,y):f(x,y)\le-1\} $ (this is not an epigraph!). Is the function $ f(x,y)=-2e^{x} +y$ convex? Is it quasiconvex? Explain based on the graph.
I know that the difference between an epigraph and a sublevel set is that an epigraph like the function is part of $\Bbb R^3$ and a sublevel set part of $\Bbb R^2$.
$ -2e^{x} +y\le-1 $
$ y\le2e^{x}-1 $
So, this is the graph of $ -2e^{x} +y=-1 $
The represented sublevel set is not convex and the function is neither convex. Is that correct?
Can I say that the function is not convex, because if
$f$ is a convex function $\Rightarrow$ it's level sets are convex?
This not here the case.
I know that if:
- $f$ is convex $\Leftarrow\Rightarrow$ $epi f$ is a convex level set
- $f$ is convex $\Rightarrow$ all $S_{k}$ sublevel sets must be convex.
Best Answer
The sublevel sets of a convex function are convex. The shaded region is the intersection of a sublevel set with a convex set; it is not convex, therefore the sublevel set is not convex, therefore the function is not convex.