  Existing convergence guarantees for the mirror descent algorithm require the
objective function to have a bounded gradient or be smooth relative to a
Legendre function. The bounded gradient and relative smoothness conditions,
however, may not hold in important applications, such as quantum state
tomography and portfolio selection. In this paper, we propose a local version
of the relative smoothness condition as a generalization of its existing global
version, and prove that under this local relative smoothness condition, the
mirror descent algorithm with Armijo line search always converges. Numerical
results showed that, therefore, the mirror descent algorithm with Armijo line
search was the fastest guaranteed-to-converge algorithm for quantum state
tomography, empirically on real data-sets.
