# How to minimize ratio of L1 and square root of L2 norms

Here is the function I want to minimize:

$$\sum_i\frac{\rho_{\tau}(1-\alpha-\pmb x_i^{\top}\pmb\beta)}{\sqrt{1+\pmb\beta^{\top}\pmb\beta}}$$

where $\alpha\in\mathbb{R}$, $\pmb\beta\in\mathbb{R}^p$ are the parameters we want to optimize over and $\tau\in(0,1)$ and the $\pmb x_i$ are given and:

$$\rho_{\tau}(x)=\tau\max(0,x)+(1-\tau)\max(-x,0)$$

It is a combination of square root of L2 terms (the denominator) and L1 terms (the numerator). It is not clear to me if this problem can be framed as a convex optimization one (I guess SOCP) and if yes how to reformulate so that it is in standard (SOCP?) form. I guess an answer to either one of these two questions is what I am looking for.

# Edit:

This is the objective function described in: He, Xuming; Liang, Hua (1997). Quantile regression estimates for a class of linear and partially linear errors-in-variables models. Ungated copy.

• I can't suggest anything myself, but isn't this a bit like quantile regression? Could you perhaps add a short description of how this objective function came about, where it came from, some context? That would make it easier for other people who have a similar problem to find this question when searching. – Kirill Dec 16 '15 at 5:34
• If you plot that function for some random data, you will see that it is non-convex. – Johan Löfberg Dec 30 '15 at 10:18
• or simply plot it for the special case $x=0$, $\alpha=0$, $\tau=1$... – Johan Löfberg Dec 30 '15 at 10:21
• @JohanLöfberg: Can you post your comment as an answer: it answers my question to my satisfaction. Also, would the non convexity also hold if the 1 on the denominator would be replaced by 0? – user189035 Dec 30 '15 at 17:14
• Then you would have 1/norm(beta) which is non-convex – Johan Löfberg Dec 30 '15 at 17:59

It is non-convex, simply study the special case you get with $x=0, \alpha=0,\tau=1$. Hence, an SOCP formulation is not possible.