1. The last sentence of [Christopher M. Bishop, *Pattern Recognition and Machine Learning*][1] Section 10.1.1 Factorized distributions on p.466, states, referring to Equation $(10.9)$, that "Convergence is guaranteed because bound is convex with respect to each of the factors $q_i(Z_i)$ ([Stephen Boyd and Lleven Vandenberghe (2004), *Convex Optimization*. Cambridge University Press][2])." 
 2. The same claim is made in [the Wikipedia entry, Variational Bayesian Methods Section 2. Mean field Approximation][3] with the same reference to the Boyd & Vandenberghe book above: "An algorithm of this sort is guaranteed to converge."

I see that $q^*_j(Z_j)$ minimizes $-\mathcal L(q)$ and thus $-\mathcal L(q^*)$ is nonincreasing with respect to the iteration on $q^*$. However, I do not see how the convexity of $-\mathcal L$ with respect to $q$ leads to the convergence of $q^*$. On the contrary, I can imagine counterexamples satisfying the conditions of decreasing functional value and convexity of the function. Neither do I know where in the referenced Boyd book to find the proof. Is there a fixed-point theorem to use?


  [1]: https://www.microsoft.com/en-us/research/uploads/prod/2006/01/Bishop-Pattern-Recognition-and-Machine-Learning-2006.pdf
  [2]: https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
  [3]: https://en.wikipedia.org/wiki/Variational_Bayesian_methods#Mean_field_approximation