  Akin to the Erd\H{o}s-Rademacher problem, Linial and Morgenstern made the
following conjecture in tournaments: for any $d\in (0,1]$, among all $n$-vertex
tournaments with $d\binom{n}{3}$ many 3-cycles, the number of 4-cycles is
asymptotically minimized by a special random blow-up of a transitive
tournament. Recently, Chan, Grzesik, Kr\'al' and Noel introduced spectrum
analysis of adjacency matrices of tournaments in this study, and confirmed this
for $d\geq 1/36$.
  In this paper, we investigate the analogous problem of minimizing the number
of cycles of a given length. We prove that for integers $\ell\not\equiv 2\mod
4$, there exists some constant $c_\ell>0$ such that if $d\geq 1-c_\ell$, then
the number of $\ell$-cycles is also asymptotically minimized by the same family
of extremal examples for $4$-cycles. In doing so, we answer a question of
Linial and Morgenstern about minimizing the $q$-norm of a probabilistic vector
with given $p$-norm for any integers $q>p>1$. For integers $\ell\equiv 2\mod
4$, however the same phenomena do not hold for $\ell$-cycles, for which we can
construct an explicit family of tournaments containing fewer $\ell$-cycles for
any given number of $3$-cycles. We conclude by proposing two conjectures on the
minimization problem for general cycles in tournaments.
