x_0+\frac {1} {x_1+x_2+x_1x_2} is convex when x_1 and x_2 are positive. But how can I add this function into CVX?

@Erling Perhaps this is another candidate for your challenge. It appears to me to be convex over the stated domain. Of course, the additive x_1 term is an inconsequential distraction.

Conic modeling challenge by https://twitter.com/e_d_andersen Do you have a nonlinear constraint that can be written in a modeling language, but can’t be represented with linear, socp, sdp, exponential cone and power cones?

By substituting new variables x=x_1+1,y= x_2+1 the main problem is to write the constraint

t\geq \frac{1}{xy-1}

where x,y>1. After a short transformation this becomes

x\geq \frac{1}{y}+\frac{1}{ty}

so the following system with geometric means should do it:

x= a+b,\quad ay\geq 1,\quad bty\geq 1.

That’s why I call @Michal_Adamaszek the wizard of conic reformulation.

@ksyxtk Use geo_mean to implement the last 2 constraints in @Michal_Adamaszek’s solution.

Thank you, Michal_Adamaszek and Mark_L_Stone, this is the answer I need.