An ergodic Markov chain with transition matrix $P$ and stationary distribution ${\pi}$ is said to be reversible if $D_{{\pi}} P = P^\top D_{{\pi}}$, where $D_{{\pi}}$ denotes the diagonal matrix with the components of ${\pi}$ on its diagonal. Reversibility is a key property in Markov chain theory, with applications ranging from computational biology to network models, such as those arising in power grid analysis.

 

In this talk, we present an approach to optimization problems over reversible Markov chains, with prescribed stationary distribution ${\pi}$. Our framework is based on Riemannian optimization over suitable manifolds of stochastic matrices associated with reversible chains, and it allows us to employ efficient and reliable Riemannian solvers. Within this framework, we address the approximation of the reversible Markov chain closest to a given one. In addition, we describe an application for minimizing Kemeny’s constant, which measures the efficiency of a Markov chain in traversing its states.