  We prove that the relative entropy of entanglement is additive when \emph{at
least one of the two states} belongs to some specific class. We show that these
classes include bipartite pure, maximally correlated, GHZ, Bell diagonal,
isotropic, and generalized Dicke states. Previously, additivity was established
only if \textit{both} states belong to the same class. Moreover, we extend
these results to entanglement monotones based on the $\alpha$-$z$ R\'enyi
relative entropy. Notably, this family of monotones includes also the
generalized robustness of entanglement and the geometric measure of
entanglement. In addition, we prove that any monotone based on a quantum
relative entropy is not additive for general states. We also compute
closed-form expressions of the monotones for bipartite pure, Bell diagonal,
isotropic, generalized Werner, generalized Dicke, and maximally correlated Bell
diagonal states. Our results rely on developing a method that allows us to
recast the initial convex optimization problem into a simpler linear one. Even
though we mostly focus on entanglement theory, we expect that some of our
technical results could be useful in investigating more general convex
optimization problems.
