I am working on a optimization-related research problem and need to know if the variance of a convex function is convex. I know this can be a little vague so I'm including a (rather formal) explanation below.
Say I have a function $v_i(x)$ where $v_i: \mathbb{R}^n \rightarrow \mathbb{R}$.
Assume that for each $i = 1...s$ the function $v_i$ can be a different convex function. Also, assume that the function has the form $v_i$ with probability $p_i$ where $\sum_{i=1}^{s}{p_i} = 1$.
Define the *expectation function as:
$E[v(x)] = \sum_{i=1}^{s}{p_i v_i(x)}$
And the *variance function as:
$Var[v(x)] = \sum_{i=1}^{s}{p_i*(v_i(x) - E[v(x)])^2}$
Assume the function $v_i$ is convex for all $i = 1...s$, then:
- Is the expectation function convex? (yes, right?)
- Is the variance function convex? (unsure)
Also, if $v_i$ is affine for all $i = 1...s$, then:
- Is the expectation function convex? (yes, right?)
- Is the variance function of a convex function convex? (if not, then what is it?)