  The semi-random graph process is a single player game in which the player is
initially presented an empty graph on $n$ vertices. In each round, a vertex $u$
is presented to the player independently and uniformly at random. The player
then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For
a fixed monotone graph property, the objective of the player is to force the
graph to satisfy this property with high probability in as few rounds as
possible.
  We focus on the problem of constructing a Hamilton cycle in as few rounds as
possible. In particular, we present a novel strategy for the player which
achieves a Hamiltonian cycle in $(2+4e^{-2}+0.07+o(1)) \, n < 2.61135 \, n$
rounds, assuming that a specific non-convex optimization problem has a negative
solution (a premise we numerically support). Assuming that this technical
condition holds, this improves upon the previously best known upper bound of $3
\, n$ rounds. We also show that the previously best lower bound of $(\ln 2 +
\ln (1+\ln 2) + o(1)) \, n$ is not tight.
