DCP for Product of Convex and Log-convex Is there a way to convert a product of a convex and a log-convex function to the DCP ruleset? Specifically, I am interested in the following functions of ${\textbf{x}}$ where ${\textbf{x}}$ and ${\textbf{y}}$ are in $\mathbb{R}^d$:
$$
{{||{\textbf{y} - \textbf{x}}||}^{p}}{\exp\left( ||{\textbf{y} - \textbf{x}}||\right)}
$$
Does there exist a simple transformation to convert this expression to a DCP for implementation in CVX in MATLAB?
As an example, I have the following problem:
$$
\text{minimize} \quad \sum_{i=1}^N 
{{||{\textbf{y}_i - \textbf{x}}||}^{p}}{\exp\left( ||{\textbf{y}_i - \textbf{x}}||\right)}
\\
s.t.  \quad {{||{\textbf{y}_i - \textbf{x}}||}^{p}}{\exp\left( ||{\textbf{y}_i - \textbf{x}}||\right)} \leq \epsilon, \quad \forall i
\\
\textbf{x} \in \mathcal{A}
$$
where $\mathcal{A}$ is a convex set. Assume $\textbf{y}_i, \textbf{x} \in \mathbb{R}^2$. Furthermore, $p \in \mathbb{R}^+$.
 A: Ok. I assume that $p \geq 1$, and I also assume that you have $x \log(x)$ and $-\log(x)$ as atoms in your DCP rule-set. CVX allows you to use them, although it only guarantees an approximation.
First, note that $\phi(t) = t^p \exp(t)$ is increasing on $t \geq 0$. Thus, we can reformulate the problem as follows over the variables $\mathbf{x}, \mathbf{y_i}, \mathbf{s}, \mathbf{t}, \mathbf{r}$:
$$
\begin{aligned}
\text{minimize} &\quad \sum_{i=1}^n s_i \\
\text{s.t.}
  &\quad t_i^p \exp(t_i) \leq s_i & \forall i \\
  &\quad \|\mathbf{y}_i - \mathbf{x}\| \leq t_i &\forall i \\
  &\quad r_i^p \exp(r_i) \leq \epsilon & \forall i \\
  &\quad \|\mathbf{y}_i - \mathbf{x}\| \leq r_i &\forall i \\
  &\quad \mathbf{x} \in \mathcal{A} \\
  &\quad \mathbf{t}, \mathbf{s}, \mathbf{r} \geq 0
\end{aligned}
$$
Now, note that $t^p \exp(t) \leq s$ if and only if $t \exp(t/p) \leq s^{1/p}$. Substituting $u_i = \exp(t_i / p),~ w_i = \exp(r_i / p)$ we obtain the following equivalent problem:
$$
\begin{aligned}
\text{minimize} &\quad \sum_{i=1}^n s_i \\
\text{s.t.}
  &\quad p u_i \log(u_i) - s_i^{1/p} \leq 0& \forall i \\
  &\quad \|\mathbf{y}_i - \mathbf{x}\| -p \log(u_i) \leq 0 &\forall i \\
  &\quad p w_i \log(w_i) - \epsilon^{1/p} \leq 0 & \forall i \\
  &\quad \|\mathbf{y}_i - \mathbf{x}\| -p \log(w_i) \leq 0 &\forall i \\
  &\quad \mathbf{x} \in \mathcal{A} \\
  &\quad \mathbf{s} \geq 0 \\
  &\quad \mathbf{u}, \mathbf{w} \geq 1
\end{aligned}
$$
This problem can be built from the DCP rule-set with the additional aproximate atoms.
