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, $p \in \mathbb{R}^+$.

  • $\begingroup$ How does this expression appear in your problem? Is it the objective function? $\endgroup$ – LinAlg Jul 3 '18 at 0:42
  • $\begingroup$ @LinAlg, Thanks for your comment. I have this expression in both the objective function and the constraints. I have edited the question to give an example of this problem. $\endgroup$ – Gourab Ghatak Jul 3 '18 at 6:35
  • $\begingroup$ There is one thing I am missing. In any DCP rule set you have atomic functions you are allowed to use. Is $x \exp(x)$ on $x\geq 0$ one of your atomic functions? In addition, what do you know about $p$? $\endgroup$ – Alex Shtof Jul 3 '18 at 8:45
  • $\begingroup$ @Alex:Thanks for your comment. In my problem, I have $p \in \mathbb{R}^+$. Furthermore, I don't think $\textbf{x}\exp(\textbf{x})$ is an atomic function, otherwise it would have been simpler I guess, to solve the problem. I might be wrong, in which case, I would be grateful if someone can point out if there is an atomic function that can be utilized here. $\endgroup$ – Gourab Ghatak Jul 3 '18 at 8:55
  • $\begingroup$ As a simple case, let us assume p = 2. $\endgroup$ – Gourab Ghatak Jul 3 '18 at 8:56

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.

  • $\begingroup$ This looks perfect. I think with these transformations, the final problem is compliant with DCP ruleset. $\textbf{x}\log\textbf{x}$ can be implemented using the function "entr" representing entropy in DCP. Thanks. $\endgroup$ – Gourab Ghatak Jul 4 '18 at 9:19

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