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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.

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    $\begingroup$ 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. $\endgroup$
    – Kirill
    Commented Dec 16, 2015 at 5:34
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    $\begingroup$ If you plot that function for some random data, you will see that it is non-convex. $\endgroup$
    – Johan Löfberg
    Commented Dec 30, 2015 at 10:18
  • $\begingroup$ or simply plot it for the special case $x=0$, $\alpha=0$, $\tau=1$... $\endgroup$
    – Johan Löfberg
    Commented Dec 30, 2015 at 10:21
  • $\begingroup$ @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? $\endgroup$
    – user189035
    Commented Dec 30, 2015 at 17:14
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    $\begingroup$ Then you would have 1/norm(beta) which is non-convex $\endgroup$
    – Johan Löfberg
    Commented Dec 30, 2015 at 17:59

1 Answer 1

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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.

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