$\log(\frac{1+x\ (a+1)}{1+x \ a})$ and its perspective $y\log(\frac{1+x/y\ (a+1)}{1+x/y \ a})$ in CVX

1

Hi,

im trying to implement in CVX the following convex function -\log(\frac{1+x\ (a+1)}{1+x \ a}) with a a postive constant, and I got an error which I think is because of the DCP ruleset. Could you please help me add this function to CVX?

The same goes for the case in which the perspective operator is applied: -y\log(\frac{1+x/y\ (a+1)}{1+x/y \ a}). How could I include this expression in CVX?

I would appreciate your help.
Thanks

2

I am not convinced that this function is convex or concave. But even if it is, I am afraid it is quite unlikely that it can be represented in CVX.

3

Thank you very much for your answer. The first expression can be shown to be convex, and the perspective operation preserves convexity [Boyd 3.2.6], therefore the second is convex too.

After some manipulation I obtained the following:

$$-y\log(\frac{1+x/y\ (a+1)}{1+x/y \ a})= y\log(\frac{y}{y+ x(a+1)}) -y\log(\frac{y}{y+ x\ a})$$

I tried introducing it in cvx as: rel_entr(y, y + x*(a+1)- rel_entr(y, y + x*a) and got the DCP error Illegal operation: {convex} - {convex}

Any ideas?

4

No, I am afraid not. Unfortunately, convexity is not sufficient to establish that a function can be expressed in CVX. CVX simply does not support arbitrary convex functions, as it uses a unique geometric representation method.

5

According to my analysis, this function is convex for x> -1/(a+1), which is I assume the branch you are interested in. Again, unfortunately, I do not believe CVX can handle it.

6

Again, thank you very much for your answer and your analysis. I am indeed interested in the branch for x>=0 with a>=0, and will try to find another way to solve.