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.
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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.