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Here's a problem which I'm trying to convert to a convex standard form:

$$\min f_0(x)=x_1+2x_2^2$$ subject to: $$x_2 \ln(x_1)\ge 0$$ $$x_1^2-x_2^2=0$$

For the equality part, it appears that I can't make it an affine function, but can make it only a function like $x_2=|x_1|$, which is not affine. But is there maybe a way to do something with these constraints in order to make this equality affine?

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From the first constraint, we need $x_1 > 0$.

If $0<x_1<1$ then $\ln(x_1)<0$, hence we have $x_2\le0$. We have $x_2 = -x_1$.

If $x_1 =1$, then $x_2=\pm 1$ and the objective value is $3$.

If $x_1 > 1$, then $\ln(x_1)>0$, hence we have $x_2 \ge 0$, we have $x_2=x_1$ and the objective value is $x_1 + 2x_1^2 > 3$.

Hence it suffices to consider

$$\min x_1+2x_2^2$$

subject to $$0<x_1 \le 1$$ $$x_2=-x_1$$

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