Let 0\leq \beta\leq 1 be an objective variable. The size of \beta is N\!\times\!N.

Now, how can I impose the following?

if \beta_{i,j}>0 then \beta_{j,i}=0

inside of CVX.

Let 0\leq \beta\leq 1 be an objective variable. The size of \beta is N\!\times\!N.

Now, how can I impose the following?

if \beta_{i,j}>0 then \beta_{j,i}=0

inside of CVX.

I think you will need to implement Big M logic (llook up at https://or.stackexchange.com/ ). And you will need a tolerance on how much an element needs to be greater than 0 to trigger the condition.