How two express this function log(1+x_1+1/x_2+...+1/x_N) in CVX?

#1

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 ?

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(Michal Adamaszek) #2

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?

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#3

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.

#4

Thanks, the methods you provide seems helpful, I will try.