In a paper I find the authors say that function \log(1+\frac 1 x_1+\frac 1 x_2+\cdots+\frac1 x_N) is convex when x_1, x_2,\cdots,x_N>0, and it can be solved by CVX.
I know \log(1+\frac 1 x_1) is convex when x_1>0. But is this function \log(1+\frac 1 x_1+\frac 1 x_2+\cdots+\frac1 x_N) convex? How to write it in CVX ?
t\geq \log(1+\frac{1}{x_1}+\cdots+\frac{1}{x_N})
can be written as
e^{-t}\leq s\leq\frac{1}{1+\frac{1}{x_1}+\cdots+\frac{1}{x_N}}
The left side is exp, the right side is the hypograph of the harmonic mean (rescaled). I don’t see that cvx has a harmonic mean atom but it definitely can be done with some quadratic cones: https://docs.mosek.com/modeling-cookbook/cqo.html#harmonic-mean
EDIT: You can also go through log-sum-exp https://docs.mosek.com/modeling-cookbook/expo.html#log-sum-inv This is probably easier in cvx but will have more exponential cones.
Do you have a link to the paper?
https://arxiv.org/abs/1705.02723
The authors said that the problem described in Eq.(28) can be solved by CVX.
The Eq.(20) \hat R_{k,m}[n] is said to be convex. So the authors obtained inequality (23) by 1st order Taylor expansion.
@whc What is your specific question? There was a question posed, and solutions provided by @Michal_Adamaszek