Based on the theory of $\varphi$-irreducible Markov chains, a
convergence proof for the CMA-ES (Covariance Matrix Adaptation
Evolution Strategy) is sketched. In the first step, the original
algorithm is reduced to the rank-one update of the covariance
matrix. A suitable Markov chain is derived and convergence of the
chain to a stationary limit distribution, the invariant measure, is
shown on a class of unimodal objective functions. The result implies
log-linear convergence to the optimum, or log-linear divergence. Even
though the simplified version of the algorithm is not recommended in
real applications, its convergence is highly relevant. Additional
components have only been expected to improve convergence behavior.
In further steps a less simplified algorithm and EDAs will be tackled.