I’m not saying it can’t be done with exponential cone, but I didn’t see how, else I would have told you. log_sum_exp does invoke the exponential cone, BTW.
Fro an example of explicit use of exponential cone construct in CVX, look at mcg’s first post in Solve optimization problems of exp function . However, any use in CVX of log, exp, entr, rel_entr, log_sum_exp will invoke the exponential cone “under the hood”, so the someone who told you might be referring to any of this. Anyhow, if the someone really does know how to input this problem into CVX, please have them or you post it here.