  Convex optimization problems arise naturally in quantum information theory,
often in terms of minimizing a convex function over a convex subset of the
space of hermitian matrices. In most cases, finding exact solutions to these
problems is usually impossible. As inspired by earlier investigations into the
relative entropy of entanglement [Phys. Rev. A 78 032310 (2008)], we introduce
a general method to solve the converse problem rather than find explicit
solutions. That is, given a matrix in a convex set, we determine a family of
convex functions that are minimized at this point. This method allows us find
explicit formulae for the relative entropy of entanglement and the Rains bound,
two well-known upper bounds on the distillable entanglement, and yields
interesting information about these quantities, such as the fact that they
coincide in the case where at least one subsystem of a multipartite state is a
qubit.
