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?
Best Answer
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 $$0<x_1 \le 1$$ $$x_2=-x_1$$