Writing geo_mean( (1+x/y) ^x )

(John) #1

Dear All,
I want to write a constraint in this form (x>0 and y>0): geo_mean( (1+x/y) ^x )
I think the objective function is convex: (1+x/y)^x
Is it possible to write this constraint in cvx?
Thanks alot

(Mark L. Stone) #2

geo_mean takes a vector as its argument. What vector is the argument of geo_mean?

Then are you claiming convexity, or concavity of that expression, and do you have a proof?

Anyhow, presuming you don’t intend to have geo_mean at all, and are interested in the convex expression, (1+x/y)^x, that can be handled as
(1+x/y)^x = exp(x*log((1+x/y))) = exp(rel_entr(x+y,y) + rel_entr(y,x+y))
where the latter makes use of the result by @Michal_Adamaszek
x*log(1+x/y) = rel_entr(x+y,y) + rel_entr(y,x+y) in your previous question Writing x*log(1+x/y) .

(John) #3

Thanks for your reply,
I checked for convexity of (1+x/y)^x and It is convex. But I don’t know if the multiplication of these functions becomes convex too or not?
I just wanna ignore successive convex approximation in my formulations, I know that we can write x*log(1+x/y) in DCP rule-set as well.

(Mark L. Stone) #4

You can use either exp(rel_entr(x+y,y) + rel_entr(y,x+y)) or rel_entr(x+y,y) + rel_entr(y,x+y), depending on which you want, and avoid CVX’s successive approximation method. To do so, install CVXQUAD https://github.com/hfawzi/cvxquad and its exponential.m replacement for CVX’s version, as described on the CVXQUAD page.

Instances of rel_entr will then be handled through CVXQUAD’s Pade approximant, which seems in my experience to be much faster and more robust than CVX’s successive approximation method,

If you actually need geo_mean, then please clarify. Note that geo_mean of a scalar is just that scalar. So geo_mean only becomes non-trivial for a vector argument.

(John) #5

Thanks alot, I appreciate it. It helped me alot!