Express a function as difference of convex functions (DC)

Question

is there a way to express the function $$1-\exp \Big( \frac{-\max(0,x)^2}{\alpha} \Big)$$ as the difference of two convex functions (DC)?

Thanks

Answer 1

Any smooth function can be decomposed into a difference of convex functions. In this case, the following should work.

We want $f(x)=g(x)-h(x)$, where $g$ and $h$ are the convex functions. Since $f$ is convex for $x < \sqrt{2 \alpha}$, and concave for $x > \sqrt {2\alpha}$, we can let $g=f$ for $x < \sqrt{2 \alpha}$, and $g$ be linear for $x > \sqrt {2\alpha}$. Then by matching up derivatives we get:

$$ g(x) = \begin{cases} 0 & x \le 0 \\ 1 - e^{-x^2/\alpha} & 0<x\le \sqrt{2\alpha} \\ 1 - e^{-2}\left(\sqrt\frac{8}{\alpha} x - 3\right) & x > \sqrt{2\alpha} \end{cases} $$

$$ h(x) = \begin{cases} 0 & x\le \sqrt{2\alpha} \\ e^{-x^2/\alpha} - e^{-2}\left(\sqrt\frac{8}{\alpha} x - 3\right) & x > \sqrt{2\alpha} \end{cases} $$