Please help me solve this expression about log function in cvx: log(1/(1+x^2))

1

please help me solve this expression in cvx: log(1/(1+x^2)) . And log(1/(1+x^2)^2)?

2

x1>= log(1/(1+x2^2));x^3>= log(1/(1+x4^2)^2)

3

I believe log(1+1/x^2) , which is convex, can be handled by using y(i)/2 in place of y(i) in the last constraint of the 1st approach of section 3.2.7 Log-sum-in of Mosek Modeling Cookbook

In this specific case, that would be:

variables t x y
t >= log_sum_exp([0;y])
x >= exp(-y/2)

As for x^3 >= log(1/(1+x^2)^2), that is non-convex because RHS - LHS is non-convex. Also, RHS is individually non-convex (x = 1 is the boundary between concavity and convexity), and LHS is individually non-concave. So, a “loser” all the way around.