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Suppose I want to minimize below objective function

$\sum | g(x_i) \cdot I_{g(x_i)<0} |^2$

i.e, the latter penalty terms like $ |g(x_i)|^2 $ are only computed when $g(x_i)<0$. $|g(x_i)|^2$ are convex functions. Is there a way to formulate it? I know sum of convex functions are convex, but if I have added this strange "filter" on the objective function, I hope it's still convex..

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    $\begingroup$ No it will not be convex. You can just look at a simple example, such as $g(x)=x^2-1$ $\endgroup$ Sep 12 at 17:58
  • $\begingroup$ What do you optimize over? The $x_i$? $\endgroup$ Sep 12 at 23:18
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I'm going to assume that you optimize over the locations $x_i$. Then this is most easily reformulated via slack variables as follows: $$ \min_{x_i,s_i} s_i^2 \\ \text{so that}\quad s_i \le g(x_i) \\ \qquad\quad s_i \le 0 $$ This is not a convex problem because the feasible region described by these constraints is not convex. You can see this by just considering the function $g(x)=x^2-1$ mentioned by @johanlofberg in a comment, for example, and plotting the feasible region spanned by the two constraints.

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