well the convergence of the global optima remains an open problem but that is more often than not the case with non-linear optimization problems.

The \epsilon term is like a leash attached to the solution point. The term adds a small multiple of divergence from a fixed point to the solution. The small multiple insures that that the objective function is not modified by much in a region around that point, but as the solution starts to diverge to infinity, the term gets large and penalizes the quality of solution.

a are not eigenvalues. They are the mixing coeffecients or "bregman projections" of the data on the basis vectors in V.
And we are not doing the minimization seperately, it is an alternating procedure, where we alternate between estimating each one of them.