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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..
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.
