DCP formulation of sum of nonconvex and convex functions

Question

I am trying to find a DCP formulation for the following convex objective function (using CVXPY):

Let $x$ be the $N$-dimensional vector variable on which we optimize on, $c$ be a known scalar value such that $0 < c \le 1$ and and $L$ be a known $N$-dimensional vector with $L_i > 0 \: \forall i=1,..., N$. The (convex) objective is formulated as:

$$ f(x) = \sum_{i=1}^N\left(c^{\max(x_i,0)/L_i}\cdot L_i - \ln(c)\cdot\max(x_i,0)\right) $$

Single out any $i=1,...,N$, the issue I am facing is that each convex element of the sum above is given by the sum of a nonconvex and a convex function:

$$ c^{\max(x_i,0)/L}\cdot L_i - \ln(c)\cdot\max(x_i,0) $$

The expected behavior of the above sum is to be equal to: \begin{align} c^{\max(x_i,0)/L}\cdot L_i - \ln(c)\cdot\max(x_i,0)=\begin{cases} c^{x_i/L}\cdot L_i - \ln(c)\cdot x&\quad x_i \ge 0 \\ L_i &\quad x_i < 0 \end{cases} \end{align}

But using standard atoms it is not trivial to provide a DCP certificate for the above due to non-convexity of the first term.

Do you have any ideas/suggestions on how one could rewrite the problem and make it DCP compliant?

Answer 1

For all $c>0$ the function of one variable $$g(x)=c^{x/L}\cdot L - \ln(c)\cdot x=\exp(\ln(c)x/L)\cdot L-\ln(c)\cdot x$$ is convex increasing on $x\geq 0$ (derivative check) and DCP representable (verbatim). So we can model your extension to all real numbers via

$$u\geq \max(x,0),$$ $$t\geq g(u).$$

Now if $x\geq 0$ then we have $t\geq g(x)$ and if $x\leq 0$ then we have $t\geq g(0)=L$.