Hi,
I have a convex optimization problem whose objective function is of the form \log(1+\dfrac{ax}{bx+c}). The constraints are all affine and x\geq 0. I am sure the problem is convex, however, I cannot find a way to implement it in CVX. I know there has been some similar questions in this forum but I was just wondering if there is any new feature or idea which helps me to write this problem in a form acceptable for CVX.
By making linear substitutions you should be able to reduce the function to \log(1+1/y) or \log(1-1/y) depending on a,b,c. The entropy section of http://docs.mosek.com/cheatsheets/conic.pdf shows how to represent them with cones. These functions also appeared on this forum in the past.
If this logarithm is the ONLY term in the objective then you can also just minimize/maximize the rational function you have inside - logarithm is monotone and the optimal point will be the same.
Thank you for your reply.
No actually the logarithm is not the only term. In fact, the objective function is a summation of the logarithms of this form where the variables are x_1, x_2,…,x_K.