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?


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.