How to espress log(1+1/x+1/y) in cvx

1

i already know how to express
log(1+1/x) = log((1+x)/x) = -log(x/(1+x)) = -log(1 - 1/(1+x)) = -log(1 - inv_pos(1+x))
but how to how to espress log(1+1/x+1/y) in cvx
help help

2

Some approaches can be found in https://docs.mosek.com/modeling-cookbook/expo.html#log-sum-inv

3

i still don’t know how to deal this problem after reading these approaches, would you do me a favor

4

I believe the first approach would be (using zx and zy in place of x_i and y_i)

variables x y t zx zy
t >= log_sum_exp([0 zx zy])
x >= exp(-zx)
y >= exp(-zy)

Then use t in place of log(1+1/x+1/y)

5

how about log(1 + a/(x + b) + a/(y + b)) I’ve tried so many times, but it doesn’t work.

6

In my post above, change
x >= exp(-zx)
to
(x+b)/a >= exp(-zx)

which is just using (x+b)/a as the “x” in that post.

Similarly for the other inequality constraint.