Let $f$ be a jointly convex function of 2 variables say $x,y$. I am interested in solving the optimization problem $$\min_{x,y\in\Delta} f(x,y)$$ where $\Delta$ is a $d$ dimensional simplex. An intuitive algorithm that I have come up with is a coordinate mirror descent algorithm, where in each iteration we perform two steps. In the first step I fix one variable, say $x$, and optimize w.r.t y (almost) exactly by performing many steps of mirror descent, with the entropy function. The second step is similar to the first step, but now we shall perform mirror descent w.r.t. the second variable. This algorithm is performed repeatedly until convergence.

My question is that have people studied such algorithms? I know that there is literature on coordinate descent, but I have not seen any coordinate mirror descent type of algorithms. I am particularly interested in convergence guarantees, and rates of convergence. Intuitions/counter-examples are also welcome.

I know that I can avoid mirror descent procedure, and instead perform gradient descent coordinate-wise. The reason why I want to avoid this is because mirror descent with the entropy function is a natural fit when the constraint set is a simplex, and I can devise simple exponential updates for each sub problem being solved.

P.S. Turns out that, such coordinate mirror descent algorithms, will converge provably. For further results see the following paper. http://arxiv.org/abs/1309.2249