Rel_entr constraint

Nonconvex
1

Dear respected members,

I am new to convex optimization, took a graduate course in optimization so I formulated some research related problems and am testing them out. So far they are working well, but I want to build on it by adding few more constraints.

I have a new constraint which is of the form (given x and y are positive):
rel_entr(x,x+y) <= (positive real constant) which is really the same as:
xlog(1+(y/x)) - (positive real constant)<=0

This is not valid for CVX as it is a concave expression on the left. Is there a way to manipulate it to be accepted by CVX? This constraint won’t make or break the overall problem but it would be very helpful to know how to manipulate it if it all possible.

Thank you very much.

Abdul

2

It appears that you have a typo. rel_entr(x,x+y) <= (positive real constant) is a convex constraint and is accepted by CVX. However, x*log(1+(y/x)) - (positive real constant) = -rel_entr(x,x+y) - (positive real constant). Therefore, apparently you meant -rel_entr(x,x+y) <= (positive real constant), and that is an inherently non-convex constraint (the inequality is going the wrong direction to be convex), for which there is no reformulation to convex constraint.

3

Thanks Mark; you are correct it was a typo. I thought as much but needed the confirmation. Thanks again; I appreciate the quick response!