Prove convexity of simple function

Question

I'm trying to understand a quiz question from CVX101, the convex optimisation MOOC.

Suppose we have $x \in \mathbb{R}^n$. We define

$$\begin{aligned} (x)_+ &= \max \{0,x\}\\ (x)_- &= \max\{0,-x\} \end{aligned}$$

such that $x = (x)_+ - (x)_-$. I want to show that $$f(x) = \mathbf{1}^Tx_- - 0.5\cdot\mathbf{1}^Tx_+$$ is a convex function.

Clearly the first part is convex (sum of convex functions), but the second part is concave. Plotting in 1D shows that it is clearly convex also, I'm just struggling to prove it.

Any help would be much appreciated.

Answer 1

Clearly the function is separable.

Let's focus on one dimensional problem.

$$f_1(x) = \max(0,-x) -0.5\max(0,x) = \max(-x, -0.5x)$$

Notice that maximum of convex function is convex.

Hence the function is convex.