  Consider the problem of minimizing a convex differentiable function on the
probability simplex, spectrahedron, or set of quantum density matrices. We
prove that the exponentiated gradient method with Armjo line search always
converges to the optimum, if the sequence of the iterates possesses a strictly
positive limit point (element-wise for the vector case, and with respect to the
Lowner partial ordering for the matrix case). To the best our knowledge, this
is the first convergence result for a mirror descent-type method that only
requires differentiability. The proof exploits self-concordant likeness of the
log-partition function, which is of independent interest.
