  Logarithmic Sobolev inequalities are a fundamental class of inequalities that
play an important role in information theory. They play a key role in
establishing concentration inequalities and in obtaining quantitative estimates
on the convergence to equilibrium of Markov processes. More recently, deep
links have been established between logarithmic Sobolev inequalities and strong
data processing inequalities. In this paper we study logarithmic Sobolev
inequalities from a computational point of view. We describe a hierarchy of
semidefinite programming relaxations which give certified lower bounds on the
logarithmic Sobolev constant of a finite Markov operator, and we prove that the
optimal values of these semidefinite programs converge to the logarithmic
Sobolev constant. Numerical experiments show that these relaxations are often
very close to the true constant even for low levels of the hierarchy. Finally,
we exploit our relaxation to obtain a sum-of-squares proof that the logarithmic
Sobolev constant is equal to half the Poincar\'e constant for the specific case
of a simple random walk on the odd $n$-cycle, with $n\in\{5,7,\dots,21\}$.
Previously this was known only for $n=5$ and even $n$.
