The function I wish to evaluate is:
$$ f(x)*t + \frac{g(x)}{t} $$
where $x\in \mathbb{R}^{n}$. $f(x)$ is a function of the variance with respect to the $n$-dimensional variable $x$:
$$ f(x)=\frac{\sum\limits_{i=1}^n (x_{i}\cdot b_{i}-\frac{1}{n}\sum\limits_{j=1}^n x_{j }\cdot b_{j})^2}{n} $$ where $b_i\geq 0$.
$g(x)$ is a linear function of $x$: $g(x) = \sum\limits_{i\in \mathcal{I}}(1-a_{i})*x_{i}, a_{i}\leq 1$. The variable $t\in \mathbb{R}^+$.
How can the convexity or concavity of this function be determined? I am able to prove that the variance function and the function $g(x)$ are convex. I have studied some basics of convex optimization, understand the definitions of convex sets, convex functions, etc., but I cannot judge the convexity of the above function. I hope to be able to prove that the above function is convex to solve it using cvxpy. Or is there a way to transform the above function into a more solvable form?
Thank you very much!
The form of your problem kind of reminded me about the "perspective" operator, which is a convex-preserving operator, but much less known than the other convex-preserving operations. With this, we can see that by g's linearity, we have that : $$\frac{g(x)}{t} = g\bigg(\frac{x}{t}\bigg) = g(P(x,t))$$
where P(.,.) is the perspective operator, and by the sets in which x and t are defined, then g(x)/t is convex with respect to (x,t). Then, it is left to show that $f(x)\cdot t$ is convex, but currently I don't know if it goes from here. At least by my knowledge, product of convex functions need not to be convex...
Look up "convex-preserving operations" for a more detailed survey of the perspective function :)